some inequalities of majorization type abstract we show some majorization inequalities and apply them to derive norm eigenvalue singular value and trace inequalities of matrices we also present a generalization and a different proof of a recent result on majorization concerning partitioned hermitian positive semidefinite matrices

normal families new perspectives this paper surveys some surprising applications of a lemma characterizing normal families of meromorphic functions on plane domains these include short and efficient proofs of generalizations of i the picard theorems ii gol dberg s theorem a meromorphic function on c which is the solution of a first order algebraic differential equation has finite order and iii the fatou julia theorem the julia set of a rational function of degree d 2 is the closure of the repelling periodic points we also discuss bloch s principle and provide simple solutions to some problems of hayman connected with this principle over twenty years ago on the way to a partial explication of the phenomenon known as bloch s principle i proved a little lemma characterizing normal families of holomorphic and meromorphic functions on plane domains 68 over the years the lemma has grown and in dextrous hands proved amazingly versatile with applications to a wide variety of topics in function theory and related areas with the renewed interest in normal families arising largely from the important role they play in complex dynamics it seems sensible to survey some of the most striking of these applications to the one variable theory with the aim of making this technique available to as broad an audience as possible that is the purpose of this report one pleasant aspect of the theory is that judicious application of the lemma often leads to proofs which seem almost magical in their brevity in such cases we have made no effort to resist the temptation to write out complete proofs hardly anything beyond a basic knowledge of function theory is required to understand what follows so the reader is urged to take courage and plough on through and now we turn to our tale 1 let d be a domain in the complex plane c we shall be concerned with analytic maps i e meromorphic functions f d r2 received by the editors october 15 1997 and in revised form may 26 1998 1991 mathematics subject classification primary 30d45 secondary 30d35 34a20 58f23

the existence of a global fundamental solution for homogeneous hormander operators via a global lifting method we prove the existence of a global fundamental solution x y with pole x for any hormander operator l i 1mxi2 on rn which is homogeneous of degree 2 here homogeneity is meant with respect to a family of non isotropic diagonal maps of the form x 1x1 nxn with 1 1 n due to a global lifting method for homogeneous operators proved by folland comm partial differential equations 2 1977 161 207 there exists a carnot group g and a polynomial surjective map g rn such that l is related to a sub laplacian lg on g we show that it is always possible to perform a global change of variable on g such that the lifting map becomes the projection of g rn rp onto rn we prove that an integration argument over the non compact fibers of provides a fundamental solution for l indeed if g x x y y x y rn x y rp is the fundamental solution of lg we show that g x 0 y y is always integrable with respect to y rp and its y integral is a fundamental solution for l

normal families of meromorphic functions concerning shared values abstract in this paper we study the problem of normal families of meromorphic functions concerning shared values and prove that a family f of meromorphic functions in a domain d is normal if for each pair of functions f and g in f f a f n and g a g n share a value b in d where n is a positive integer and a b are two finite constants such that n 4 and a 0 this result is not true when n 3

large sets at infinity and maximum principle on unbounded domains for a class of sub elliptic operators abstract maximum principles on unbounded domains play a crucial role in several problems related to linear second order pdes of elliptic and parabolic type in this paper we consider a class of sub elliptic operators l in r n and we establish some criteria for an unbounded open set to be a maximum principle set for l we extend some classical results related to the laplacian by deny hayman and kennedy and to the sub laplacians on stratified lie groups by bonfiglioli and the second named author

a cutting surface method for uncertain linear programs with polyhedral stochastic dominance constraints in this paper we study linear optimization problems with a newly introduced concept of multidimensional polyhedral linear second order stochastic dominance constraints by using the polyhedral properties of this dominance condition we present a cutting surface algorithm and show its finite convergence the cut generation problem is a difference of convex functions dc optimization problem we exploit the polyhedral structure of this problem to present a novel branch and cut algorithm that incorporates concepts from concave minimization and binary integer programming a linear programming problem is formulated for generating concavity cuts in our case where the polyhedra are unbounded we also present duality results for this problem relating the dual multipliers to utility functions without the need to impose constraint qualifications which again is possible because of the polyhedral nature of the problem numerical examples are presented showing the nature of solutions of our model a corrected pdf has been appended to the original

exceptional functions and normal families of meromorphic functions with multiple zeros abstract let k be a positive integer with k 2 let h 0 be a holomorphic function which has no simple zeros in d and let f be a family of meromorphic functions defined in d all of whose poles are multiple and all of whose zeros have multiplicity at least k 1 if for each function f f f k z h z then f is normal in d

characterization of the hilbert ball by its automorphism group let be a bounded convex domain in a separable hilbert space the authors prove a version of the theorem of bun wong which asserts that if such a domain admits an automorphism orbit accumulating at a strongly pseudoconvex boundary point then it is biholomorphic to the ball key ingredients in the proof are a new localization argument using holomorphic peaking functions and the use of new normal families arguments in the construction of the limit biholomorphism

composition factors of functors abstract let x be a family of finite groups satisfying certain conditions and k be a field we study composition factors radicals and socles of biset and related functors defined on x over k for such a functor m and for a group h in x we construct bijections between some classes of maximal respectively simple subfunctors of m and some classes of maximal respectively simple k out h submodules of m h we use these bijections to relate the multiplicity of a simple functor s h v in m to the multiplicity of v in a certain k out h module related to m h we then use these general results to study the structure of one of the important biset and related functors namely the burnside functor b k which assigns to each group g in x its burnside algebra b k g k z b g where b g is the burnside ring of g we find the radical and the socle of b k in most cases of x and k for example if k is of characteristic p 0 and x is a family of finite abelian p groups we find the radical and the socle series of b k considered as a biset functor on x over k we finally study restrictions of functors to nonfull subcategories for example we find some conditions forcing a simple deflation functor to remain simple as a mackey functor for an inflation functor m defined on abelian groups over a field of characteristic zero we also obtain a criterion for m to be semisimple in terms of the images of inflation and induction maps on m

some remarks on g functors and the brauer morphism keywords finite group g functor restriction maps transfer maps mackey s formula frobenius reciprocity brauer morphism defect base nilpotent ideal burnside ring idempotent formula induction formula symmetric g functor character ring green ring reference ctg article 1988 003 record created on 2008 12 16 modified on 2017 05 12

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

optimization with multivariate stochastic dominance constraints we consider stochastic optimization problems where risk aversion is expressed by a stochastic ordering constraint the constraint requires that a random vector depending on our decisions stochastically dominates a given benchmark random vector we identify a suitable multivariate stochastic order and describe its generator in terms of von neumann morgenstern utility functions we develop necessary and sufficient conditions of optimality and duality relations for optimization problems with this constraint assuming convexity we show that the lagrange multipliers corresponding to dominance constraints are elements of the generator of this order thus refining and generalizing earlier results for optimization under univariate stochastic dominance constraints furthermore we obtain necessary conditions of optimality for non convex problems under additional smoothness assumptions

stochastic dominance constrained markov decision processes we are interested in risk constraints for infinite horizon discrete time markov decision processes mdps starting with average reward mdps we show that increasing concave stochastic dominance constraints on the empirical distribution of reward lead to linear constraints on occupation measures an optimal policy for the resulting class of dominance constrained mdps is obtained by solving a linear program we compute the dual of this linear program to obtain average dynamic programming optimality equations that reflect the dominance constraint in particular a new pricing term appears in the optimality equations corresponding to the dominance constraint we show that many types of stochastic orders can be used in place of the increasing concave stochastic order we also carry out a parallel development for discounted reward mdps with stochastic dominance constraints a portfolio optimization example is used to motivate the paper

from stochastic dominance to mean risk models semideviations as risk measures abstract two methods are frequently used for modeling the choice among uncertain outcomes stochastic dominance and mean risk approaches the former is based on an axiomatic model of risk averse preferences but does not provide a convenient computational recipe the latter quantifies the problem in a lucid form of two criteria with possible trade off analysis but cannot model all risk averse preferences in particular if variance is used as a measure of risk the resulting mean variance markowitz model is in general not consistent with stochastic dominance rules this paper shows that the standard semideviation square root of the semivariance as the risk measure makes the mean risk model consistent with the second degree stochastic dominance provided that the trade off coefficient is bounded by a certain constant similar results are obtained for the absolute semideviation and for the absolute and standard deviations in the case of symmetric or bounded distributions in the analysis we use a new tool the outcome risk o r diagram which appears to be particularly useful for comparing uncertain outcomes

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

subharmonic functions on carnot groups many results of classical potential theory are extended to sub laplacians on carnot groups some characterizations of subharmonicity representation formulas of poisson jensen s kind and nevanlinna type theorems are proved we also characterize the riesz measure related to bounded above subharmonic functions in n

optimization with stochastic dominance constraints we introduce stochastic optimization problems involving stochastic dominance constraints we develop necessary and sufficient conditions of optimality and duality theory for these models and show that the lagrange multipliers corresponding to dominance constraints are concave nondecreasing utility functions the models and results are illustrated on a portfolio optimization problem

cut generation for optimization problems with multivariate risk constraints we consider a class of stochastic optimization problems that features benchmarking preference relations among random vectors representing multiple random performance measures criteria of interest given a benchmark random performance vector preference relations are incorporated into the model as constraints which require the decision based random vector to be preferred to the benchmark according to a relation based on multivariate conditional value at risk cvar or second order stochastic dominance ssd we develop alternative mixed integer programming formulations and solution methods for cut generation problems arising in optimization under such multivariate risk constraints the cut generation problems for cvar and ssd based models involve the epigraphs of two distinct piecewise linear concave functions which we refer to as reverse concave sets we give the complete linear description of the linearization polytopes of these two non convex substructures we present computational results that show the effectiveness of our proposed models and methods

normal families of meromorphic functions with multiple values let f be a family of meromorphic functions defined in a domain d let 0 be a holomorphic function in d and k be a positive integer suppose that for every function f f f 0 f k 0 and all zeros of f k z have multiplicities at least k 2 k if for k 1 has only zeros with multiplicities at most 2 and for k 2 has only simple zeros then f is normal in d this improves and generalizes the related results of gu fang and chang yang schwick et al

normal families of functions for subelliptic operators and the theorems of montel and koebe abstract a classical theorem of montel states that a family of holomorphic functions on a domain c uniformly bounded on the compact subsets of is a normal family the aim of this paper is to obtain a generalization of this result in the subelliptic setting of families of solutions u to l u 0 where l belongs to a wide class of real divergence form pdos comprising sub laplacians on carnot groups subelliptic laplacians on arbitrary lie groups as well as the laplace beltrami operator on riemannian manifolds to this end we extend another remarkable result due to koebe we characterize the solutions to l u 0 as fixed points of suitable mean value operators with non trivial kernels a suitable substitute for the cauchy integral formula is also provided finally the local boundedness assumption is relaxed by replacing it with l loc 1 boundedness

eigenvalue problems and fixed point theorems for a class of positive nonlinear operators we study the eigenvalue problems for a class of positive nonlinear operators defined on a cone in a banach space using projective metric techniques and schauder s fixed point theorem we establish existence uniqueness monotonicity and continuity results for the eigensolutions moreover the method leads to a result on the existence of a unique fixed point of the operator applications to nonlinear boundary value problems to differential delay equations and to matrix equations are considered

inequalities theory of majorization and its applications by albert w marshall and ingram olkin abstract in the years since the appearance of marshall and olkin s masterful survey the influence of this work has been so great that some comments however incomplete and idiosyncratic on subsequent developments will be of value figure 1 an appendix lists minor errata to the first printing of the book

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

normal families and shared values of meromorphic functions abstract let k 2 be a positive integer let f be a family of meromorphic functions in a domain d all of whose zeros have multiplicity at least k 1 and let a z 0 h z 0 be two holomorphic functions on d if for each f f f a z f k h z then f is normal in d

normal families and value distribution in connection with composite functions abstract we prove a value distribution result which has several interesting corollaries let k n let c and let f be a transcendental entire function with order less than 1 2 then for every nonconstant entire function g we have that f g k has infinitely many zeros this result also holds when k 1 for every transcendental entire function g we also prove the following result for normal families let k n let f be a transcendental entire function with f 1 k and let a 0 a k 1 a be analytic functions in a domain then the family of analytic functions g such that f g k z j 0 k 1 a j z f g j z a z in is a normal family

optimality and duality theory for stochastic optimization problems with nonlinear dominance constraints we consider a new class of optimization problems involving stochastic dominance constraints of second order we develop a new splitting approach to these models optimality conditions and duality theory these results are used to construct special decomposition methods

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