probabilistic generation of finite simple groups ii abstract in earlier work it was shown that each nonabelian finite simple group g has a conjugacy class c such that whenever 1 x g the probability is greater than 1 10 that g x y for a random y c much stronger asymptotic results were also proved here we show that allowing equality the bound 1 10 can be replaced by 13 42 and excluding an explicitly listed set of simple groups the bound 2 3 holds we use these results to show that any nonabelian finite simple group g has a conjugacy class c such that if x 1 x 2 are nontrivial elements of g then there exists y c such that g x 1 y x 2 y similarly aside from one infinite family and a small explicit finite set of simple groups g has a conjugacy class c such that if x 1 x 2 x 3 are nontrivial elements of g then there exists y c such that g x 1 y x 2 y x 3 y we also prove analogous but weaker results for almost simple groups

generation and random generation from simple groups to maximal subgroups abstract let g be a finite group and let d g be the minimal number of generators for g it is well known that d g 2 for all non abelian finite simple groups we prove that d h 4 for any maximal subgroup h of a finite simple group and that this bound is best possible we also investigate the random generation of maximal subgroups of simple and almost simple groups by applying a recent theorem of jaikin zapirain and pyber we show that the expected number of random elements generating such a subgroup is bounded by an absolute constant we then apply our results to the study of permutation groups in particular we show that if g is a finite primitive permutation group with point stabilizer h then d g 1 d h d g 4

nodal type bound states of schrodinger equations via invariant set and minimax methods for nonlinear schrodinger equations in the entire space we present new results on invariant sets of the gradient flows of the corresponding variational functionals the structure of the invariant sets will be built into minimax procedures to construct nodal type bound state solutions of nonlinear schrodinger type equations 2004 elsevier inc all rights reserved

a characterization of the finite simple groups abstract in this paper we obtain a quantitative characterization of all finite simple groups let t g denote the set of indices of maximal subgroups of group g and let p g be the smallest number in t g we have the following theorems theorem 2 let n and g be finite simple groups if n divides g p n p g and t n t g then or and 2 11 or 2 and 3 theorem 3 let n be a finite simple and let g be a finite group if g n and t g t n then g n

a class of p q laplacian type equation with concave convex nonlinearities in bounded domain abstract in this paper our main purpose is to establish the existence of multiple solutions of a class of p q laplacian equation involving concave convex nonlinearities p u q u v x u r 2 u u p 2 u f x u x u 0 x where is a bounded domain in r n 0 1 r q p n and p n p n p is the critical sobolev exponent s u div u s 2 u is the s laplacian of u we prove that for any 0 0 is a constant there is a 0 such that for every 0 the above problem possesses infinitely many weak solutions we also obtain some results for the case 1 q p r p the existence results of solutions are obtained by variational methods

simple irreducible subgroups of exceptional algebraic groups a closed subgroup of a semisimple algebraic group is called irreducible if it lies in no proper parabolic subgroup in this paper we classify all irreducible subgroups of exceptional algebraic groups g which are connected closed and simple of rank at least 2 consequences are given concerning the representations of such subgroups on various g modules for example with one exception the conjugacy classes of irreducible simple connected subgroups of rank at least 2 are determined by their composition factors on the adjoint module of g

constant sign and nodal solutions of coercive p q laplacian problems abstract the existence of a greatest negative a smallest positive and a nodal weak solution to a dirichlet problem with p q laplacian and reaction term that makes coercive the corresponding energy functional is investigated via variational methods besides truncation techniques

regularity for double phase variational problems we prove sharp regularity theorems for minimisers of a class of variational integrals whose integrand switches between two different types of degenerate elliptic phases according to the zero set of a modulating coefficient a cdot the model case is given by the functional w mapsto int dw p a x dw q rm d x where q p and a cdot geqq 0

radical subgroups of the finite exceptional groups of lie type e6 abstract we consider the finite exceptional groups of lie type e 6 1 q e 6 q and e 6 1 q e 6 2 q both the universal versions we classify up to conjugacy the maximal p local subgroups and radical p subgroups of g e 6 e q for p 5 with p q and q e mod p and for p 3 with 3 q and q e mod 3 as an application the essential p rank of the frobenius category f d g is determined where d is a sylow p subgroup of g moreover if p 3 then we show that there is a subgroup h f 4 q of g containing d such that f d g f d h that is h controls 3 fusion in g

on ground states of superlinear p laplacian equations in rn abstract we consider superlinear p laplacian equations in r n with a potential which is periodic or has a bounded potential well without assuming the ambrosetti rabinowitz type condition and the monotonicity of the function t f x t t p 1 we prove the existence of ground states even for the case p 2 our results extend the recent results of li wang and zeng ann inst h poincare anal non lineaire 23 2006 829 837

multiplicity of positive solutions for a fractional p q laplacian problem in rn abstract in this paper we deal with the following fractional p q laplacian problem p s u q s u v e x u p 2 u u q 2 u f u in r n u w s p r n w s q r n u 0 in r n where s 0 1 e 0 is a small parameter 2 p q n s t s with t p q is the fractional s t laplacian operator v r n r is a continuous function satisfying the global rabinowitz condition and f r r is a continuous function with subcritical growth using suitable variational arguments and ljusternik schnirelmann category theory we prove that the above problem admits multiple solutions for e 0 small enough

the existence of a nontrivial solution to the p q laplacian problem with nonlinearity asymptotic to up 1 at infinity in rn abstract we consider the following elliptic problem p u m u p 2 u q u n u q 2 u f x u x r n u w 1 p r n w 1 q r n where m n 0 n 3 and 1 q p n f x u u p 1 tends to a positive constant l as u we prove in this paper that the problem possesses a nontrivial solution even if the nonlinearity f x t does not satisfy the ambrosetti rabinowitz condition 0 f x u 0 u f x s d s 1 p f x u u for x u r n r

nonlocal harnack inequalities abstract we state and prove a general harnack inequality for minimizers of nonlocal possibly degenerate integro differential operators whose model is the fractional p laplacian

subgroups of maximal rank in finite exceptional groups of lie type the purpose of this paper is to investigate a large and natural class of maximal subgroups of the finite exceptional groups of lie type which we call subgroups of maximal rank these subgroups play a prominent role both in the classification of local maximal subgroups in 9 theorem 1 and in the reduction theorem for subgroups in 28 theorem 2 in 9 it is shown that any local maximal subgroup of a finite exceptional group of lie type is either a subgroup of maximal rank or one of a small list of exceptions and in 28 using this result it is proved that any maximal subgroup is either of maximal rank or almost simple or in a known list to describe these subgroups we require the following notation let g be a simple adjoint algebraic group over an algebraically closed field k of characteristic p 0 and let o be an endomorphism of g such that l ga is a finite exceptional simple group of lie type over fq where q p let a be a group such that f x l the group aut l is generated by ga together with field and graph automorphisms see 5 38 all of which extend to morphisms of the abstract group g commuting with o thus there is a subgroup x of caut c a such that x x o and so x acts on the set of a stable subsets of g for a a stable subset y we write nx y for the stabilizer in x of y if d is a a stable closed connected reductive subgroup of g containing a maximal torus t of g and m nx d we call m a subgroup of maximal rank in x in this paper we determine the structure and conjugacy classes of those subgroups of maximal rank which are maximal in x

global gradient estimates for the borderline case of double phase problems with bmo coefficients in nonsmooth domains abstract we consider a double phase problem with bmo coefficient in divergence form on a bounded nonsmooth domain the problem under consideration is characterized by the fact that both ellipticity and growth switch between a type of polynomial and a type of logarithm according to the position which describes a feature of strongly anisotropic materials we obtain the global calderon zygmund type estimates for the distributional solution in the case that the associated nonlinearity has a small bmo and the boundary of the domain is sufficiently flat in the reifenberg sense

existence of a nontrivial solution for the p q laplacian in rn without the ambrosetti rabinowitz condition abstract in this paper we prove the existence of at least one nonnegative nontrivial weak solution in d 1 p r n d 1 q r n for the equation p u q u a x u p 2 u b x u q 2 u f x u x r n where 1 q p q n q n q p n m u div u m 2 u is the m laplacian operator the coefficients a and b are continuous coercive and positive functions and the nonlinearity f is a caratheodory function satisfying some hypotheses which do not include the ambrosetti rabinowitz condition

p q systems with critical terms in rn abstract this paper deals with the existence of nontrivial solutions for critical hardy quasilinear systems s driven by general p q elliptic operators of marcellini types existence is derived as an application of the concentration compactness principle of lions via the mountain pass geometry the constructed solution has both components nontrivial that is it solves the actual system which does not reduce into an equation we present also a simplified version s of the main system s which is anyway interesting we exhibit a new proof for the existence of nontrivial solutions of s which is more direct and elegant however the assumptions for both systems s and s are milder and in any case much different from the usual requests granted in related problems finally the results improve or complement previous theorems for the quasilinear p q scalar as well as vectorial problems

regularity theory for fully nonlinear integro differential equations we consider nonlinear integro differential equations like the ones that arise from stochastic control problems with purely jump levy processes we obtain a nonlocal version of the abp estimate harnack inequality and interior c 1 regularity for general fully nonlinear integro differential equations our estimates remain uniform as the degree of the equation approaches two so they can be seen as a natural extension of the regularity theory for elliptic partial differential equations

hybrid systems with memory existence and well posedness of generalized solutions hybrid systems with memory are dynamical systems exhibiting both hybrid and delay phenomena while systems of this type are frequently encountered in many physical and engineering systems particularly in control applications various issues centered around the robustness of hybrid delay systems have not been adequately dealt with in this paper we establish some basic results on a framework that allows us to study hybrid systems with memory through generalized concepts of solutions in particular we develop the basic existence of generalized solutions using regularity conditions on the hybrid data which are formulated in a phase space of hybrid trajectories equipped with the graphical convergence topology in contrast with the uniform convergence topology that has often been used adopting the graphical convergence topology allows us to establish well posedness of hybrid systems with memory we then show that as a consequence of well posedness preasymptotic stability of well posed hybrid systems wit

on perturbations of a class of a periodic m laplacian equation with critical growth 1 after the well known results of brezis nirenberg 10 problems involving ellipticequationswithcriticalgrowthinboundeddomainshavebeenstudiedbyseveralauthors see for example 7 15 11 13 17 and the references therein the special features of this class of problems considered in this paper is that it isdened in r

on bruck loops of 2 power exponent the goal of this paper is two fold first we provide the information needed to study bol ar or bruck loops by applying group theoretic methods this information is used in this paper as well as in baumeister and stein 2010 bs3 and in stein 2009 s moreover we determine the groups associated to bruck loops of 2 power exponent under the assumption that every non abelian simple group s is either passive or isomorphic to psl2 q q 1 4 a 2 power in a separate paper it is proven that indeed every non abelian simple group s is either passive or isomorphic to psl2 q q 1 4 a 2 power stein 2009 s the results obtained here are used in baumeister and stein 2010 bs3 where we determine the structure of the groups associated to the finite bruck loops

existence of multiple positive solutions for a p q laplacian system with critical nonlinearities abstract in this paper our main purpose is to establish the existence of multiple positive solutions of a p q laplacian system with convex and sobolev critical nonlinearities by some standard variational methods whose key is to construct homotopies between and levels of the functional j and some analytical techniques

factorizations of simple algebraic groups we determine all factorizations of simple algebraic groups as the product of two maximal closed connected subgroups additional results are established which drop the maximality assumption and applications are given to the study of subgroups of classical groups transitive on subspaces of a given dimension

on conjugacy classes of maximal subgroups of finite simple groups and a related zeta function we prove that the number of conjugacy classes of maximal subgroups of bounded order in a finite group of lie type of bounded rank is bounded for exceptional groups this solves a longstanding open problem the proof uses among other tools some methods from geometric invariant theory using this result we provide a sharp bound for the total number of conjugacy classes of maximal subgroups of lie type groups of fixed rank drawing conclusions regarding the behaviour of the corresponding zeta function g s m maxg g m s which appears in many probabilistic applications more specifically we are able to show that for simple groups g and for any fixed real number s 1 g s 0 as g this confirms a conjecture made in 27 we also apply these results to prove the conjecture made in 29 that the symmetric group sn has n o 1 conjugacy classes of primitive maximal subgroups 2000 mathematics classification numbers 20e28 20g15 20d06

cyclic quotients of transitive groups abstract let a be a transitive subgroup of s n we show that the largest cyclic quotient of a has order at most n this can be interpreted as an equivalent result about extensions of constants in the galois closure of a covering of curves over a finite field we also prove that the point stabilizer in a finite primitive permutation group always has a faithful orbit

sharp regularity for functionals with p q growth we prove regularity theorems for minimizers of integral functionals of the calculus of variations f x du x dx with non standard growth conditions of p q type z p f x z l z q 1 p q in particular we find that a sufficient condition for minimizers to be regular is qp n n where the function f x z is holder continuous with respect to the x variable and x rn this condition is also sharp we include results in the setting of orlicz spaces moreover we treat certain relaxed functionals too finally we address a problem posed by marcellini in 43 showing a minimizer with an isolated singularity

harnack inequalities for double phase functionals we prove a harnack inequality for minimisers of a class of non autonomous functionals with non standard growth conditions they are characterised by the fact that their energy density switches between two types of different degenerate phases

large subgroups of simple groups abstract let g be a finite group a proper subgroup h of g is said to be large if the order of h satisfies the bound h 3 g in this note we determine all the large maximal subgroups of finite simple groups and we establish an analogous result for simple algebraic groups in this context largeness is defined in terms of dimension an application to triple factorisations of simple groups both finite and algebraic is discussed

razumikhin type theorems for hybrid system with memory in this paper we address the stability problem of hybrid systems with memory which are newly developed to model hybrid dynamic systems affected by delays for such systems stability criteria are developed based on razumikhin type conditions which are usually applied in the delay systems we prove that the hybrid system with memory is uniformly globally pre asymptotically stable by small gain theorem besides a relaxed razumikhin type stability criterion is also proposed which does not require strict decrease of the lyapunov razumikhin function during jumps several examples are given to illustrate the effectiveness and generality of the proposed results

existence of infinitely many solutions for p laplacian equations in rn abstract based on the cohomological linking method for cones and a new direct sum decomposition of w 1 p r n we study the existence and multiplicity of solutions of a p laplacian equation in r n with sign changing potential and subcritical p superlinear nonlinearity our assumptions are mild and different from those studied previously

nonlinear calderon zygmund theory in the limiting case we prove a maximal differentiability and regularity result for solutions to nonlinear measure data problems specifically we deal with the limiting case of the classical theory of calderon and zygmund in the setting of nonlinear possibly degenerate equations and we show a complete linearization effect with respect to the differentiability of solutions a prototype of the results obtained here tells for instance that if rm div du p 2 du mu quad mbox in omega subset mathbb r n with mu being a borel measure with locally finite mass on the open subset omega subset mathbb r n and p 2 1 n then du p 2 du in w sigma 1 rm loc omega quad mbox for every sigma in 0 1 the case sigma 1 is obviously forbidden already in the classical linear case of the poisson equation triangle u mu

on the stationary solutions of generalized reaction diffusion equations with p q laplacian in this paper we analyse a family of stationary nonlinear equations with p q laplacian delta p u delta q u lambda c x u which have a wide spectrum of applications in many areas of science we introduce a new type of variational principles corresponding to this family of equations using this formalism we exhibit intervals for the scalar parameter lambda where there exist positive solutions of the considered problems furthermore we prove in another interval the nonexistence of nontrivial solutions these results are different from those of existence and nonexistence for stationary equations with single laplacian

multiplicity of solutions for a quasilinear elliptic problem via the cohomological index abstract in this paper we study a quasilinear elliptic problem with the dirichlet boundary condition in a bounded domain involving the operator a u p u q u assuming that the nonlinearity has a concave convex behavior we obtain some multiplicity results more precisely we obtain five nontrivial solutions two by a minimization argument two by the mountain pass theorem and the other by a cohomological linking theorem

regularity and existence of solutions of elliptic equations with p q growth conditions for some positive constants m a4 and for exponents q 2 p 2 2 under 1 2 1 3 and some other assumptions by assuming also that the quotient q p is sufficiently close to one in dependence on n precisely if q p c n n 2 then we prove that every weak solution to 1 1 of class 4 4 0 is locally lipschitz continuous in q moreover there are positive constants 8 c and 0 2 1 such that

primitive permutation groups with a solvable 2 transitive subconstituent for a permutation group g acting on a finite set and a point a suborbit is an orbit of the point stabilizer g on the permutation group induced by g on is called a subconstituent of g moreover g is said to be uniprimitive if g is primitive but not 2 transitive in this paper we investigate uniprimitive permutation groups which have a solvable 2 transitive subconstituent we determine all such groups g which have a simple socle the affine case that is g has an elementary abelian socle are also discussed and an infinite family of affine primitive groups with non self paired 2 transitive subconstituents are presented

base sizes for simple groups and a conjecture of cameron let g be a permutation group on a finite set a base for g is a subset b with pointwise stabilizer in g that is trivial we write b g for the smallest size of a base for g in this paper we prove that b g 6 if g is an almost simple group of exceptional lie type and is a primitive faithful g set an important consequence of this result when combined with other recent work is that b g 7 for any almost simple group g in a non standard action proving a conjecture of cameron the proof is probabilistic and uses bounds on fixed point ratios

irreducible a1 subgroups of exceptional algebraic groups abstract a closed subgroup of a semisimple algebraic group is called irreducible if it lies in no proper parabolic subgroup in this paper we classify all irreducible a 1 subgroups of exceptional algebraic groups g consequences are given concerning the representations of such subgroups on various g modules for example the conjugacy classes of irreducible a 1 subgroups are determined by their composition factors on the adjoint module of g

holder regularity for nonlocal double phase equations abstract we prove some regularity estimates for viscosity solutions to a class of possible degenerate and singular integro differential equations whose leading operator switches between two different types of fractional elliptic phases according to the zero set of a modulating coefficient a a the model case is driven by the following nonlocal double phase operator u x u y p 2 u x u y x y n s p d y a x y u x u y q 2 u x u y x y n t q d y where q p and a 0 our results do also apply for inhomogeneous equations for very general classes of measurable kernels by simply assuming the boundedness of the modulating coefficient we are able to prove that the solutions are holder continuous whereas similar sharp results for the classical local case do require a to be holder continuous to our knowledge this is the first regularity result for nonlocal double phase problems

wang s multiplicity result for superlinear equations without the ambrosetti rabinowitz condition we consider a nonlinear elliptic equation driven by the sum of a p laplacian and a q laplacian where 1 q 2 p with a p 1 superlinear caratheodory reaction term which doesn t satisfy the usual ambrosetti rabinowitz condition using variational methods based on critical point theory together with techniques from morse theory we show that the problem has at leat three nontrivial solutions among them one is positive and one is negative

the existence of a positive solution to asymptotically linear scalar field equations which is important in using the mountain pass theorem the aim of this paper is to discuss how to use the mountain pass theorem to show the existence of a positive solution to the present problem when we lose the above condition furthermore if f x u f u we also prove that the above problem has a ground state by using the articial constraint method

bounded minimisers of double phase variational integrals bounded minimisers of the functional w mapsto int dw p a x dw q rm d x where 0 leqq a cdot in c 0 alpha and 1 p q are c 1 beta regular provided the sharp bound q leqq p alpha holds

lyapunov based sufficient conditions for stability of hybrid systems with memory hybrid systems with memory are dynamical systems exhibiting both hybrid and delay phenomena in this note we study the asymptotic stability of hybrid systems with memory using generalized concepts of solutions these generalized solutions motivated by studying robustness and well posedness of such systems are defined on hybrid time domains and parameterized by both continuous and discrete time we establish lyapunov based sufficient conditions for asymptotic stability using both lyapunov razumikhin functions and lyapunov krasovskii functionals examples are provided to illustrate these conditions

on the number of classes of conjugate hall subgroups in finite simple groups abstract in this paper we find the number of classes of conjugate hall subgroups in all finite almost simple groups we also complete the classification of hall subgroups in finite simple groups and correct some mistakes from our previous paper

nonlocal curvature flows this paper aims at building a unified framework to deal with a wide class of local and nonlocal translation invariant geometric flows we introduce a class of nonlocal generalized mean curvatures and prove the existence and uniqueness for the level set formulation of the corresponding geometric flows we then introduce a class of generalized perimeters whose first variation is an admissible generalized curvature within this class we implement a minimizing movements scheme and we prove that it approximates the viscosity solution of the corresponding level set pde we also describe several examples and applications besides recovering and presenting in a unified way existence uniqueness and approximation results for several geometric motions already studied and scattered in the literature the theory developed in this paper also allows us to establish new results

existence of solutions for quasilinear elliptic equations abstract using variational methods we study the existence and multiplicity of solutions of the dirichlet problem for the equation div a u p u p 2 u f x u

some classes of eigenvalues problems for generalized p q laplacian type operators on bounded domains abstract the aim of this paper is to present a preliminary study on a more general class of p q type eigenvalues problems given by div a u p u p 2 u u m 2 u in u 0 on where is a smooth bounded domain in r n n 3 2 p n m r with m 1 in suitable ranges listed later a is a c 1 real function and 0 is a real parameter we make use of variational approaches like a generalized version of weierstrass theorem and or the ekeland variational principle and the mountain pass theorem due to ambrosetti rabinowitz in particular the existence of a continuous and unbounded set of positive generalized eigenvalues is established

existence of nonnegative solutions for quasilinear elliptic equations with indefinite critical nonlinearities abstract we study the quasilinear elliptic equation div u u a x f u b x g u in with dirichlet boundary condition u 0 on where is a bounded domain in r n a x b x are sign changing continuous functions and g u has critical growth at infinity with respect to the principal part a nonnegative nontrivial solution is given under appropriate growth conditions on f u g u at 0 and

multiplicity of positive solutions for a p q laplacian system with concave and critical nonlinearities abstract in this paper we consider a p q laplacian system with concave and sobolev critical nonlinearities by using the n ehari manifold fibering maps and the lusternik schnirelman category we prove that the problem has at least cat 1 distinct positive solutions where cat denotes the lusternik schnirelman category of in itself

existence of positive solutions for a class of p q elliptic problems with critical growth on rn this paper is concerned with the existence of positive solutions to the class of p q elliptic problems with critical growth type div a u p u p 2 u b u p u p 2u f u u 2u u z 0 x rn where is a positive parameter a r r is a function of c1 class and b f r r are continuous functions

nonlinear commutators for the fractional p laplacian and applications we prove a nonlinear commutator estimate concerning the transfer of derivatives onto testfunctions for the fractional p laplacian this implies that solutions to certain degenerate nonlocal equations are higher differentiable also weakly fractional p harmonic functions which a priori are less regular than variational solutions are in fact classical as an application we show that sequences of uniformly bounded frac n s harmonic maps converge strongly outside at most finitely many points

radical 3 subgroups of f4 q with q even abstract in a recent paper an and huang classified the radical 3 subgroups of f 4 q for q odd here we use a similar approach to classify the radical 3 subgroups of f 4 q with q even as an application we determine the essential rank of the frobenius category f d f 4 q for q even where d is a sylow 3 subgroup of f 4 q

lyapunov conditions for input to state stability of hybrid systems with memory this paper studies input to state stability for hybrid systems with memory which models hybrid dynamics affected by time delays using both lyapunov razumikhin functions and lyapunov krasovskii functionals lyapunov based sufficient conditions are established for input to state stability in addition further extensions and relaxations are proposed for special cases such as the stable flow jump cases and the cases that lyapunov functions do not decrease strictly during flow jumps finally two examples are used to illustrate the developed results

multiplicity of positive solutions to a p q laplacian equation involving critical nonlinearity abstract in this paper our main purpose is to establish the existence of multiple solutions of a p q laplacian equation with convex and sobolev critical nonlinearity by some standard variational methods whose key is to construct homotopies between and levels of the functional i and some analytical techniques

nonlocal tug of war and the infinity fractional laplacian motivated by the tug of war game studied by peres et al in 2009 we consider a nonlocal version of the game that goes as follows at every step two players pick respectively a direction and then instead of flipping a coin in order to decide which direction to choose and then moving a fixed amount e 0 as is done in the classical case it is an s stable levy process that chooses at the same time both the direction and the distance to travel starting from this game we heuristically derive a deterministic nonlocal integrodifferential equation that we call the infinity fractional laplacian we study existence uniqueness and regularity both for the dirichlet problem and for a double obstacle problem both problems having a natural interpretation as tug of war games 2011 wiley periodicals inc

maximal 2 local subgroups of e7 q abstract let g e 7 q be the finite exceptional group of lie type universal version we classify up to conjugacy all maximal proper 2 local subgroups of g that is all 2 local m g which are maximal with respect to inclusion among all proper subgroups of g which are 2 local for this purpose we first determine up to conjugacy all elementary abelian 2 subgroups of g containing the center z g these classifications are an important first step towards a classification of the 2 radical subgroups of g

irreducible disconnected subgroups of exceptional algebraic groups this dissertation is concerned with the study of irreducible embeddings of simple algebraic groups of exceptional type it is motivated by the role of such embeddings in the study of positive dimensional closed subgroups of classical algebraic groups the classification of the maximal closed connected subgroups of simple algebraic groups was carried out by e b dynkin g m seitz and d m testerman their analysis for the classical groups was based primarily on a striking result if g is a simple algebraic group and o g sl v is a tensor indecomposable irreducible rational representation then with specified exceptions the image of g is maximal among closed connected subgroups of one of the classical groups sl v sp v or so v in the case of closed not necessarily connected subgroups of the classical groups one is interested in considering irreducible embeddings of simple algebraic groups and their automorphism groups given a simple algebraic group y defined over an algebraically closed field k one is led to study the embeddings g 0 g is a closed non connected positive dimensional subgroup of y and v is a nontrivial irreducible rational ky module such that v g is irreducible we obtain a precise description of such triples g y v

higher sobolev regularity for the fractional p laplace equation in the superquadratic case abstract we prove that for p 2 solutions of equations modeled by the fractional p laplacian improve their regularity on the scale of fractional sobolev spaces moreover under certain precise conditions they are in w loc 1 p and their gradients are in a fractional sobolev space as well the relevant estimates are stable as the fractional order of differentiation s reaches 1

on a class of superlinear p q laplacian type equations on rn starting from a new sum decomposition of w1 p rn w1 q rn and using a variational approach we investigate the existence of multiple weak solutions of a p q laplacian equation on rn for 1 q p n with a sign changing potential and a caratheodory reaction term satisfying the celebrated ambrosetti rabinowitz condition our assumptions are mild and different from those used in related papers and moreover our results improve or complement previous ones for the single p laplacian

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

existence of nontrivial solutions for p laplacian equations in rn abstract in this paper we consider a p laplacian equation in r n with sign changing potential and subcritical p superlinear nonlinearity by using the cohomological linking method for cones an existence result is obtained we also give a result on the existence of periodic solutions for one dimensional p laplacian equations which can be proved by the same method

hybrid systems with delayed jumps asymptotic stability via robustness and lyapunov conditions hybrid systems subject to delayed jumps form a class of dynamical systems with broad applications this article develops sufficient conditions for robust asymptotic stability of hybrid systems in the presence of delayed jumps more precisely given a delay free hybrid system we introduce a higher order delayed system parametrized by the length of delays we show that when the delay parameter is set to zero the higher order model captures the solutions of the delay free system under mild conditions it is shown that when the delay free system has an asymptotically stable compact set for small enough delays solutions of the delayed system converge to a neighborhood of a set of interest related to the aforementioned compact set then lyapunov functions for the delay free system are used to develop sufficient conditions for asymptotic stability in the presence of delays unlike prior work in the literature the results pertaining to these notions of stability hold for systems possessing zeno solutions with time varying delays importantly the required conditions are expressed in finite dimensional space and depend primarily on the data of the delay free system the practical stability result is validated numerically through the hybrid system model of a controlled boost converter circuit with state triggered switches and zeno solutions the higher order model and the derived lyapunov conditions are utilized to obtain quantitative bounds on maximum allowable delays for switched systems and sampled data control

existence of least energy positive negative and nodal solutions for a class of p q problems with potentials vanishing at infinity abstract in this paper we prove the existence of at least three nontrivial solutions for the class of p q problems given by div a u p u p 2 u v x b u p u p 2 u k x f u in r n where n 3 2 p n a b f are c 1 real functions and v k are positive and continuous functions thanks to a hardy type inequality we are able to prove compact embeddings in weighted sobolev spaces and by means of mountain pass theorem we get a positive and a negative ground state weak solution moreover by a minimization argument and a quantitative deformation lemma we find a nodal or sign changing weak solution with two nodal domains or that changes sign exactly once in r n the functions f a and b satisfy suitable growth and monotonicity conditions which allow this result to extend the previous ones to a larger class of p q type problems