gravity coupled with matter and the foundation of non commutative geometry we first exhibit in the commutative case the simple algebraic relations between the algebra of functions on a manifold and its infinitesimal length elementds its unitary representations correspond to riemannian metrics and spin structure whileds is the dirac propagatords x x d 1 whered is the dirac operator we extend these simple relations to the non commutative case using tomita s involutionj we then write a spectral action the trace of a function of the length element which when applied to the non commutative geometry of the standard model will be shown cc to give the sm lagrangian coupled to gravity the internal fluctuations of the non commutative geometry are trivial in the commutative case but yield the full bosonic sector of sm with all correct quantum numbers in this slightly non commutative case the group of local gauge transformations appears spontaneously as a normal subgroup of the diffeomorphism group

local superlinearity and sublinearity for the p laplacian we study the existence nonexistence and multiplicity of positive solutions for a family of problems updelta p u f lambda x u u in w 0 1 p upomega where is a bounded domain in mathbb r n n p and 0 is a parameter the family we consider includes the well known nonlinearities of ambrosetti brezis cerami type in a more general form namely lambda a x u q b x u r where 0 leqslant q p 1 r leqslant p 1 here the coefficient a x is assumed to be nonnegative but b x is allowed to change sign even in the critical case preliminary results of independent interest include the extension to the p laplacian context of the brezis nirenberg result on local minimization in w 0 1 p text and c 0 1 a c 1 alpha estimate for equations of the form updelta p u h x u with h of critical growth a strong comparison result for the p laplacian and a variational approach to the method of upper lower solutions for the p laplacian

c 1 conformal field theories on riemann surfaces we study the theory ofc 1 torus and 2 orbifold models on general riemann surfaces the operator content and occurrence of multi critical points in this class of theories is discussed the partition functions and correlation functions of vertex operators and twist fields are calculated using the theory of double covered riemann surfaces it is shown that orbifold partition functions are sensitive to the torelli group we give an algebraic construction of the operator formulation of these nonchiral theories on higher genus surfaces modular transformations are naturally incorporated as canonical transformations in the hilbert space

observing the state of a synchronous generator part 2 applications new portional and proportional integral observers that estimate the transient state of a synchronous generator are presented the transient state model of a generator is represented by a set of non linear differential equations observers for four synchronous generators are given digital computer simulation for three types of generators s 3125 mva 432 mva and 555 mva confirms that the non linear observers estimate the state correctly not only in a local region of the equilibrium point but also in a large area which includes the typical regime of generator transient processes

some qualitative questions on the equation div a x u u f x u abstract in this article we establish several applications of picone s identity for the operator of the form d i v a x u u such as hardy type inequality sturmian comparison theorem monotonicity property of the first eigenvalue nonexistence of positive supersolutions and caccioppoli inequality under certain conditions on a

multiple positive solutions for quasilinear problems with indefinite sublinear nonlinearity abstract we have established multiplicity of nontrivial solutions for the quasilinear elliptic problem p u h x u 1 g x u in u 0 in u 0 on where r n is a smooth bounded domain p 1 1 p h is a measurable function and g r r is a caratheodory function such that g x 0 0 which is asymptotically linear we also prove existence and nonexistence results when g x t k x t p 1

results for degenerate nonlinear elliptic equations involving a hardy potential abstract consider the problem p div a x u u u s x p f x in u x 0 in u x 0 on where is an open bounded subset of r n n 3 1 p n and s are positive numbers f is a nonnegative function in some lebesgue space a r r n r n is such that c 0 a x t p 1 p a x t for some 0 1 which provides a noncoercive operator when u the problem could be seen as a reaction model which produces a saturation effect that is the diffusion goes to zero when u goes to infinity this type of reaction appears as a linearization of the arrhenius reaction in some solid combustion problems see for instance peral and vazquez 1995 21 the aim of the article is to study existence nonexistence and regularity of the solutions to problem p the main features in the paper are the following the threshold on s to have existence or nonexistence of solution in some weak sense is s p 1 1 moreover a bound on is needed to have solution in the critical case s p 1 1 the regularity of the solution according to the summability of f

limits and degenerations of unitary conformal field theories in the present paper degeneration phenomena in conformal field theories are studied for this purpose a notion of convergent sequences of cfts is introduced properties of the resulting limit structure are used to associate geometric degenerations to degenerating sequences of cfts which as familiar from large volume limits of non linear sigma models can be regarded as commutative degenerations of the corresponding quantum geometries as an application the large level limit of the a series of unitary virasoro minimal models is investigated in detail in particular its geometric interpretation is determined

wzw fusion rings in the limit of infinite level we show that the wzw fusion rings at finite levels form a projective system with respect to the partial ordering provided by divisibility of the height i e the level shifted by a constant from this projective system we obtain wzw fusion rings in the limit of infinite level this projective limit constitutes a mathematically well defined prescription for the classical limit of wzw theories which replaces the naive idea of sending the level to infinity the projective limit can be endowed with a natural topology which plays an important role for studying its structure the representation theory of the limit can be worked out by considering the associated fusion algebra this way we obtain in particular an analogue of the verlinde formula

computing the first eigenvalue of the p laplacian via the inverse power method abstract in this paper we discuss a new method for computing the first dirichlet eigenvalue of the p laplacian inspired by the inverse power method in finite dimensional linear algebra the iterative technique is independent of the particular method used in solving the p laplacian equation and therefore can be made as efficient as the latter the method is validated theoretically for any ball in r n if p 1 and for any bounded domain in the particular case p 2 for p 2 the method is validated numerically for the square

on the antimaximum principle for the discrete p laplacian with sign changing weight abstract this work deals with the antimaximum principle for the discrete neumann and dirichlet problem p u k 1 m k u k p 2 u k h k in 1 n we prove the existence of three real numbers 0 a b the problem has no solution moreover these three real numbers are optimal

weighted eigenvalue problems for quasilinear elliptic operators with mixed robin dirichlet boundary conditions abstract we investigate the existence of principal eigenvalues type problems with weights for the quasilinear operator p v p with mixed weighted robin dirichlet boundary conditions in a bounded regular domain we also give some results on the existence of nonprincipal eigenvalues

sturmian comparison theorem for half linear second order differential equations let be defined by s s p 2 s with p 1 a fixed number we extend sturm comparison theorem of the linear differential equation to the nonlinear differential equation by using the wirtinger inequality a lyapunov inequality and some oscillation criteria of e are also given

bifurcation problems for the laplacian in ⁿ in this paper we consider the bifurcation problem div ivulp 2vu ag x julp 2u f a x u in rn with p 1 we show that a continuum of positive solutions bifurcates out from the principal eigenvalue a1 of the problem div ivuip 2vu ag x ulp 2u here both functions f and g may change sign

global attractors and convergence to equilibrium for degenerate ginzburg landau and parabolic equations abstract we study the dynamics of a degenerate complex ginzburg landau cgl and a real parabolic equation with a variable generally nonsmooth diffusion coefficient which may vanish at some points or be unbounded for the cgl equation there exists a global attractor in l 2 for the parabolic equation we show the existence of a global branch of nonnegative stationary states the global bifurcation result is used in order to establish in conjunction with the definition of a gradient dynamical system in the natural phase space that any solution with nonnegative initial data tends to the trivial or the nonnegative equilibrium

existence of minimal solutions to quasilinear elliptic equations with several sub natural growth terms abstract we study the existence of positive solutions to quasilinear elliptic equations of the type p u u q in r n in the sub natural growth case 0 q p 1 where p u u p 2 u is the p laplacian with 1 p n and and are nonnegative radon measures on r n we construct minimal generalized solutions under certain generalized energy conditions on and to prove this we give new estimates for interaction between measures we also construct solutions to equations with several sub natural growth terms using the same methods

on the principal eigenvalue of a second order linear elliptic problem on considere le probleme de dirichlet 7b lu mu dans u 0 sur dans un domaine borne r n a frontiere lisse 7b l est un operateur differentiel uniformement elliptique sur de la forme 7b lu i n j 1 a ij 2 u x j x i i n 1 a i u x i a 0 u a coefficients a valeurs reels appartenant a c 0 1

non existence and existence of entire solutions for a quasi linear problem with singular and super linear terms abstract we establish results concerning non existence and existence of entire positive solutions for the nonlinear elliptic problem p u a x u m b x u n in r n u 0 in r n u x x 0 where m p 1 n with 1 p n a b 0 a b 0 are locally holder continuous functions and 0 is a real parameter the main purpose of this paper in short is to complement the principal theorem of b xu and z yang entire bounded solutions for a class of quasi linear elliptic equations boundary value problems 2007 art id 16407 8 pp showing existence and non existence of solutions for the above problem for 0 appropriately

very fast linear and non linear observers part 1 theory the problem of the design of very fast observers is considered for linear and nonlinear systems four optimal criteria are introduced allowing a wide choice of observer parameters and an allocation of desired eigenvalues newly derived approaches are compared with known methods of optimal observer design numerical examples illustrate these methods

on the principal eigenvalues of indefinite elliptic problems on considere les deux systemes d equations lu u 2 b j d j u au wu lv wv k v dans un domaine borne lisse r n on suppose que a w sont des matrices m m ueh 1 2 on donne des conditions suffisantes pour l existence de vecteurs propres positifs et des bornes superieures et inferieures sur les valeurs propres correspondantes

antimaximum principle in rn local versus global we study the antimaximum principle for the p laplacian defined on the whole set rn with an indefinite weight function we show that a local version of this principle holds but that the global version does not hold

semilinear biharmonic problems with a singular term abstract the aim of this work is to study the optimal exponent p to have solvability of problem 2 u u x 4 u p c f in u 0 in u u 0 on where p 1 0 c 0 and r n n 4 is a smooth and bounded domain such that 0 we consider f 0 under some hypothesis that we will precise later

generalized minimizer solutions for equations with the p laplacian and a potential term let varomega be a domain in mathbb r n possibly unbounded n geq2 1 and let v in l mathrm loc infty varomega consider the energy functional mathcal q v on c mathrm c infty varomega and its g teaux derivative mathcal q v prime respectively given by mathcal q v u eqdef frac 1 p int varomega nabla u p v u p mathrm d x qquad mathcal q v prime u mathrm div nabla u p 2 nabla u v u p 2 u for u in c mathrm c infty varomega assume that mathcal q v 0 on c mathrm c infty varomega setminus 0 and that mathcal q v does not have a ground state in the sense of a null sequence for mathcal q v that converges in l mathrm loc p varomega to a positive function varphi in c mathrm loc 1 varomega a ground state finally let f in mathcal d prime varomega be such that the functional u mapsto mathcal q v u langle u f rangle c mathrm c infty varomega to mathbb r is bounded from below then the equation mathcal q v prime u f has a solution u 0 in w 1 p mathrm loc varomega in the sense of distributions this solution also minimizes the functional u mapsto mathcal q v u langle u f rangle c mathrm c infty varomega to mathbb r where mathcal q v denotes the bipolar vargamma regularization of mathcal q v and mathcal q v is the largest convex weakly lower semicontinuous functional on c mathrm c infty varomega that satisfies mathcal q v leq mathcal q v the original energy functional mathcal q v is not necessarily convex

constant sign and nodal solutions for a class of nonlinear dirichlet problems abstract we consider a nonlinear dirichlet problem with a caratheodory reaction which has arbitrary growth from below we show that the problem has at least three nontrivial smooth solutions two of constant sign and the third nodal in the semilinear case i e p 2 with the reaction f z being c 1 and with subcritical growth we show that there is a second nodal solution for a total of four nontrivial smooth solutions finally when the reaction has concave terms and is subcritical and for the nonlinear problem i e 1 p we show that again we can have the existence of three nontrivial smooth solutions two of constant sign and a third nodal

caffarelli kohn nirenberg type equations of fourth order with the critical exponent and rellich potential abstract we study the existence nonexistence of positive solutions of 2 u u x 4 u q 2 u x in where is a bounded domain and n 5 q 2 n n 4 0 4 and 0 n n 4 4 2 we prove the nonexistence result when is an open subset of r n which is star shaped with respect to the origin we also study the existence of positive solutions when is a smooth bounded domain with a nontrivial topology and 0 0 0 for certain 0 n n 4 4 2 and n 8 different behaviors are obtained for palais smale sequences depending on whether 0 or 0

observing the state of a synchronous generator part 1 theory the problem of observing the state in non linear systems is presented a new approach to the problem of non linear observer stability is developed this approach thanks to the application of massera s theorem allows the construction of non linear observers for a wide class of systems and the proof of their stability the method for gain matrix k selection are just the same as for linear observers a new concept of non linear proportional integral observers is given and a simple algorithm for gain element selection is developed

minimization of eigenvalues for a quasilinear elliptic neumann problem with indefinite weight we investigate the minimization of the positive principal eigenvalue of the problem pu m u p 2u in u 0 on over a class of sign changing weights m with m 0 it is proved that minimizers exist and satisfy a bang bang type property in dimension one we obtain a complete description of the minimizers this problem is motivated by applications from population dynamics

canonical form observer design for non linear time variable systems an observer of canonical phase variable form for non linear time variable systems is introduced the development of this non linear time variable form requires regularity of the non linear time variable observability matrix of the system from the relationships derived during the development it follows that a non linear time variable observer can be dimensioned by an eigenvalue assignment with respect to the canonical state coordinates if a linearization of system non linearities about the reconstructed state trajectories is permissible this is an assumption similar to that of the extended kalman filter based on a linearization about the current estimate

the geometry of the limit of n 2 minimal models we consider the limit of two dimensional n 2 2 superconformal minimal models when the central charge approaches c 3 starting from a geometric description as nonlinear sigma models we show that one can obtain two different limit theories one is the free theory of two bosons and two fermions the other one is a continuous orbifold thereof we substantiate this claim by detailed conformal field theory computations

the existence and non existence of positive solutions to a singular quasilinear elliptic problem in rn abstract this paper deals with the existence and nonexistence of entire positive solutions of the quasilinear elliptic equation p u a x g u b x f u in r n 1 p n n 3 with u x 0 as x here either g or f or both of them are singular at 0 in the sense that g t f t as t 0 by using a perturbation method which eliminates the singularity on a ball with radius r and then let r tends to infinity with help of the bounded super solution of the original problem we obtain the existence of a weak solution of the problem the main results of this paper improve the corresponding results of dragos patru covei the existence and asymptotic behavior of a positive solution to a quasilinear elliptic problem in r n nonlinear anal 69 2008 2615 2622 j v goncalves c a santos singular elliptic problems existence non existence and boundary behavior nonlinear anal 66 2007 2078 2090

fredholm alternative for the p laplacian in higher dimensions abstract in this paper we characterize the set of all right hand sides h c formula for which the boundary value problem p u 1 u p 2 u h in u 0 on has at least one weak solution u w01 p here 1 0 is the first eigenvalue of the p laplacian in particular we prove that for h 1 0 this problem is solvable and the energy functional e h u 1 p u p 1 p u p hu is unbounded from below

necessary and sufficient conditions of existence for a system involving the p laplacian 1 p n in this work we study the existence of positive solutions for the system p pu f x u v inrn pv g x u v inrn lim x u x 0 lim x v x where pu div u p 2 u is the p laplacian 1 in some case we can have because these parameters depend on p n and the growth of f and g the existence result is related to a sub super solution methods and the nonexistence result is a consequence of picone s identity

positive solutions of the p laplace equation with singular nonlinearity abstract for a given bounded domain in r n with c 1 boundary for some 0 1 and a possibly singular nonlinearity f on 0 we give sufficient conditions on f so that the p laplace equation p u f x u admits a solution u w 0 1 p on the basis of a comparison principle we will give a sufficient condition under which such a problem admits a unique solution

first robin eigenvalue of the p laplacian on riemannian manifolds we consider the first robin eigenvalue lambda p m alpha for the p laplacian on a compact riemannian manifold m with nonempty smooth boundary with alpha in mathbb r being the robin parameter firstly we prove eigenvalue comparison theorems of cheng type for lambda p m alpha secondly when alpha 0 we establish sharp lower bound of lambda p m alpha in terms of dimension inradius ricci curvature lower bound and boundary mean curvature lower bound via comparison with an associated one dimensional eigenvalue problem the lower bound becomes an upper bound when alpha 0 our results cover corresponding comparison theorems for the first dirichlet eigenvalue of the p laplacian when letting alpha rightarrow infty

neumann eigenvalue problems on the exterior domains abstract for p 1 we consider the following weighted neumann eigenvalue problem on b 1 c the exterior of the closed unit ball in r n 0 1 p g p 2 in b 1 c 0 on b 1 where p is the p laplace operator and g l l o c 1 b 1 c is an indefinite weight function depending on the values of p and the dimension n we take g in certain lorentz spaces or weighted lebesgue spaces and show that 0 1 admits an unbounded sequence of positive eigenvalues that includes a unique principal eigenvalue for this purpose we establish the compact embeddings of w 1 p b 1 c into l p b 1 c g for g in certain weighted lebesgue spaces for n p we also provide an alternate proof for the embedding of w 1 p b 1 c into the lorentz space l p p b 1 c further we show that the set of all eigenvalues of 0 1 is closed

the maximum principle and the existence of principal eigenvalues for some linear weighted boundary value problems abstract in this work we deal with the problem of the existence and uniqueness of principal eigenvalues for some linear weighted boundary value problems associated to a general second order uniformly elliptic operator for a large class of sign definited weights we characterize whether the boundary value problem admits a principal eigenvalue or not

a remark on the uniqueness of positive solutions for some dirichlet problems abstract in this paper we prove the uniqueness of positive weak solutions to the problem p u g x u x u 0 x by a direct argument here is a bounded domain in r n n 3 with smooth boundary p denotes the p laplacian operator defined by p z div z p 2 z p 1 g x 0 0 for a e x and g x u is a caratheodory function we provide a direct uniqueness proof for this problem under certain conditions

existence uniqueness and qualitative properties of positive solutions of quasilinear elliptic equations we study the following quasilinear elliptic equation p pu x a x up 1 b x g u 0in rn where p 1 a b l rn b g 0 b 0 and lloc rn infrn we provide a sharp criterion in term of generalized principal eigenvalues for existence nonexistence of positive solution of p in suitable classes of functions uniqueness result for p in those classes is also derived under additional conditions on we further show that i either for every 0 nonexistence phenomenon occurs ii or there exists a threshold value 0 in the sense that for every 0 existence and uniqueness phenomenon occurs and for every nonexistence phenomenon occurs in the latter case we study the limits as 0 and of the sequence of positive solutions of p our results are new even in the case p 2

collapsing riemannian manifolds while keeping their curvature bounded i soit y n une variete de riemann connexe complete et p y n le rayon d injectivite i p de l application exponentielle en p est defini comme etant le plus petit r tel que exp p b r p n est pas un diffeomorphisme sur son image on etudie la situation dans laquelle la taille du rayon d injectivite est petite relativement a la courbure

principal eigenvalues with indefinite weight functions both existence and non existence results for principal eigenvalues of an elliptic operator with indefinite weight function have been proved the existence of a continuous family of principal eigenvalues is demonstrated