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

generalized eigenvalue problems for p q laplacian with indefinite weight abstract this paper presents existence and non existence results on a positive solution for quasilinear elliptic equations of the form r u r u m r x u r 2 u in with 1 r r and 0 under dirichlet boundary condition where is a bounded domain in r n and m r is a weight function in l admitting sign change we show that existence and non existence of a positive solution depend only on the relation between and the first eigenvalue of r laplacian with weight function m r whence it is independent of the operator r and the parameter 0

projective techniques and functional integration for gauge theories a general framework for integration over certain infinite dimensional spaces is first developed using projective limits of a projective family of compact hausdorff spaces the procedure is then applied to gauge theories to carry out integration over the non linear infinite dimensional spaces of connections modulo gauge transformations this method of evaluating functional integrals can be used either in the euclidean path integral approach or the lorentzian canonical approach a number of measures discussed are diffeomorphism invariant and therefore of interest to the connection dynamics version of quantum general relativity the account is pedagogical in particular prior knowledge of projective techniques is not assumed

a multiwave approximate riemann solver for ideal mhd based on relaxation ii numerical implementation with 3 and 5 waves in the first part of this work bouchut et al j comput phys 108 7 41 2007 we introduced an approximate riemann solver for one dimensional ideal mhd derived from a relaxation system we gave sufficient conditions for the solver to satisfy discrete entropy inequalities and to preserve positivity of density and internal energy in this paper we consider the practical implementation and derive explicit wave speed estimates satisfying the stability conditions of bouchut et al j comput phys 108 7 41 2007 we present a 3 wave solver that well resolves fast waves and material contacts and a 5 wave solver that accurately resolves the cases when two eigenvalues coincide a full 7 wave solver which is highly accurate on all types of waves will be described in a follow up paper we test the solvers on one dimensional shock tube data and smooth shear waves

continuity of states on non unital differential algebras in loop quantum cosmology in a recent paper engle et al commun math phys 354 231 246 2017 showed that there is a unique state on the reduced holonomy flux algebra of homogeneous isotropic loop quantum cosmology that is invariant under residual diffeomorphims this result has been claimed to be true both for the ashtekar bojowald lewandowski framework and for that introduced by the present author unfortunately the uniqueness proof relies on an incorrect argument which spoils the second case in our short note we are going to patch this issue this way keeping the nice uniqueness result in both cases moreover we will even extend the underlying operator algebraic statements as this might help later for studying higher dimensional models

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

a multiwave approximate riemann solver for ideal mhd based on relaxation i theoretical framework we present a relaxation system for ideal magnetohydrodynamics mhd that is an extension of the suliciu relaxation system for the euler equations of gas dynamics from it one can derive approximate riemann solvers with three or seven waves that generalize the hllc solver for gas dynamics under some subcharacteristic conditions the solvers satisfy discrete entropy inequalities and preserve positivity of density and internal energy the subcharacteristic conditions are nonlinear constraints on the relaxation parameters relating them to the initial states and the intermediate states of the approximate riemann solver itself the 7 wave version of the solver is able to resolve exactly all material and alfven isolated contact discontinuities practical considerations and numerical results will be provided in another paper

quantization of diffeomorphism invariant theories of connections with a non compact structure group an example a simple diffeomorphism invariant theory of connections with the non compact structure group mathbb r of real numbers is quantized the theory is defined on a four dimensional space time by an action resembling closely the self dual plebanski action for general relativity the space of quantum states is constructed by means of projective techniques by kijowski 1 except for this point the applied quantization procedure is based on loop quantum gravity methods

some results concerning the representation theory of the algebra underlying loop quantum gravity important characteristics of the loop approach to quantum gravity are a specific choice of the algebra a of kinematical observables and of a representation of a on a measure space over the space of generalized connections this representation is singled out by its elegance and diffeomorphism covariance recently in the context of the quest for semiclassical states states of the theory in which the quantum gravitational field is close to some classical geometry it was realized that it might also be worthwhile to study different representations of the algebra a the content of the present work is the observation that under some mild assumptions the mathematical structure of representations of a can be analyzed rather effortlessly to a certain extent each representation can be labeled by sets of functions and measures on the space of generalized connections that fulfill certain conditions

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 the structure of the observable algebra of qcd on the lattice we prove that the matter field subalgebra of the observable algebra for qed on a finite lattice is isomorphic to the enveloping algebra of the lie algebra sl 2n c factorized by a certain ideal using this result we give a new proof of the decomposition of the physical hilbert space into charge superselection sectors

semiclassical limits of extended racah coefficients we explore the geometry and asymptotics of extended racah coefficients the extension is shown to have a simple relationship to the racah coefficients for the positive discrete unitary representation series of su 1 1 which is explicitly defined moreover it is found that this extension may be geometrically identified with two types of lorentzian tetrahedra for which all the faces are timelike the asymptotic formulas derived for the extension are found to have a similar form to the standard ponzano regge asymptotic formulas for the su 2 6j symbol and so should be viable for use in a state sum for three dimensional lorentzian quantum gravity

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

on the support of the ashtekar lewandowski measure we show that the ashtekar isham extension overline a g of the configuration space of yang mills theories a g is topologically and measure theoretically the projective limit of a family of finite dimensional spaces associated with arbitrary finite lattices

a fully well balanced positive and entropy satisfying godunov type method for the shallow water equations this work is devoted to the derivation of a fully well balanced numerical scheme for the well known shallow water model during the last two decades several well balanced strategies have been introduced with a special attention to the exact capture of the stationary states associated with the so called lake at rest by fully well balanced we mean here that the proposed godunov type method is also able to preserve stationary states with non zero velocity the numerical procedure is shown to preserve the positiveness of the water height and satisfies a discrete entropy inequality

uniqueness of diffeomorphism invariant states on holonomy flux algebras loop quantum gravity is an approach to quantum gravity that starts from the hamiltonian formulation in terms of a connection and its canonical conjugate quantization proceeds in the spirit of dirac first one defines an algebra of basic kinematical observables and represents it through operators on a suitable hilbert space in a second step one implements the constraints the main result of the paper concerns the representation theory of the kinematical algebra we show that there is only one cyclic representation invariant under spatial diffeomorphisms while this result is particularly important for loop quantum gravity we are rather general the precise definition of the abstract algebra of the basic kinematical observables we give could be used for any theory in which the configuration variable is a connection with a compact structure group the variables are constructed from the holonomy map and from the fluxes of the momentum conjugate to the connection the uniqueness result is relevant for any such theory invariant under spatial diffeomorphisms or being a part of a diffeomorphism invariant theory

on the solutions of generalized discrete poisson equation the set of common numerical and analytical problems is introduced in the form of the generalized multidimensional discrete poisson equation it is shown that its solutions with square summable discrete derivatives are unique up to a constant the proof uses the fourier transform as the main tool the necessary condition for the existence of the solution is provided

spin networks for noncompact groups spin networks are a natural generalization of wilson loop functionals they have been extensively studied in the case where the gauge group is compact and it has been shown that they naturally form a basis of gauge invariant observables physically the restriction to compact gauge groups is enough for the study of yang mills theories however it is well known that noncompact groups naturally arise as internal gauge groups for lorentzian gravity models in this context a proper construction of gauge invariant observables is needed the purpose of the present work is to define the notion of spin network states for noncompact groups we first build by a careful gauge fixing procedure a natural measure and a hilbert space structure on the space of gauge invariant graph connections spin networks are then defined as generalized eigenvectors of a complete set of hermitic commuting operators we show how the delicate issue of taking the quotient of a space by noncompact groups can be address in term of algebraic geometry we finally construct the full hilbert space containing all spin network states having in mind applications to gravity we illustrate our results for the groups sl 2 r and sl 2 c

quantization of diffeomorphism invariant theories of connections with local degrees of freedom quantization of diffeomorphism invariant theories of connections is studied and the quantum diffeomorphism constraint is solved the space of solutions is equipped with an inner product that is shown to satisfy the physical reality conditions this provides in particular a quantization of the husain kucha model the main results also pave the way to quantization of other diffeomorphism invariant theories such as general relativity in the riemannian case i e signature the approach appears to contain all the necessary ingredients already in the lorentzian case it will have to be combined in an appropriate fashion with a coherent state transform to incorporate complex connections

entropy satisfying flux vector splittings and kinetic bgk models we establish forward and backward relations between entropy satisfying bgk relaxation models such as those introduced previously by the author and the first order flux vector splitting numerical methods for general systems of conservation laws classically to a kinetic bgk model that is compatible with some family of entropies we can associate an entropy flux vector splitting we prove that the converse is true any entropy flux vector splitting can be interpreted by a kinetic model and we obtain an explicit characterization of entropy satisfying flux vector splitting schemes we deduce a new proof of discrete entropy inequalities under a sharp cfl condition that generalizes the monotonicity criterion in the scalar case in particular this gives a stability condition for numerical kinetic methods with noncompact velocity support a new interpretation of general kinetic schemes is also provided via approximate riemann solvers we deduce the construction of finite velocity relaxation systems for gas dynamics and obtain a hllc scheme for which we are able to prove positiveness of density and internal energy and discrete entropy inequalities

coherent state transforms for spaces of connections abstract the segal bargmann transform plays an important role in quantum theories of linear fields recently hall obtained a non linear analog of this transform for quantum mechanics on lie groups given a compact connected lie groupgwith its normalized haar measure h the hall transform is an isometric isomorphism froml2 g h to h g c l2 g c whereg c the complexification ofg h g c the space of holomorphic functions ong c and an appropriate heat kernel measure ong c we extend the hall transform to the infinite dimensional context of non abelian gauge theories by replacing the lie groupgby a certain extension of the space a g of connections modulo gauge transformations the resulting coherent state transform provides a holomorphic representation of the holonomyc algebra of real gauge fields this representation is expected to play a key role in a non perturbative canonical approach to quantum gravity in 4 dimensions

multiplicity of positive solutions to a p laplacian equation involving critical nonlinearity abstract in this paper we deal with multiplicity of positive solutions to the p laplacian equation of the type div u p 2 u u q 1 u p 1 u w 1 p 0 where r n is a bounded domain n p2 2 p q p np n p and is a positive parameter we prove that there exists 0 such that for each 0 the equation has at least cat positive solutions

multiplicity of positive solutions for some quasilinear elliptic equation in rnwith critical sobolev exponent consider the equation pu g x u p 2u f x u p 2u 1 in rn where 1 0 be the principal eigenvalue of pu g x u p 2uinrn rng x u p 0 2 withu 1 0 the associated eigenfunction we prove i equation 1 has at least one positive solution if 0 1 ii suppose rn f x u 1 p 1such that 1 has at least one positive solution for 1 0 moreover ifp 2 there exists 0 1such that 1 has at least two positive solutions for 1 0

when do measures on the space of connections support the triad operators of loop quantum gravity in this work we investigate the question under what conditions hilbert spaces that are induced by measures on the space of generalized connections carry a representation of certain non abelian analogues of the electric flux we give the problem a precise mathematical formulation and start its investigation for the technically simple case of u 1 as gauge group we establish a number of no go theorems asserting that for certain classes of measures the flux operators can not be represented on the corresponding hilbert spaces the flux observables we consider play an important role in loop quantum gravity since they can be defined without recurse to a background geometry and they might also be of interest in the general context of quantization of non abelian gauge theories

combined effects of concave and convex nonlinearities in some elliptic problems abstract this paper deals with a class of semilinear elliptic dirichlet boundary value problems where the combined effects of a sublinear and a superlinear term allow us to establish some existence and multiplicity results

separable hilbert space in loop quantum gravity we study the separability of the state space of loop quantum gravity in the standard construction the kinematical hilbert space of the diffeomorphism invariant states is nonseparable this is a consequence of the fact that the knot space of the equivalence classes of graphs under diffeomorphisms is noncountable however the continuous moduli labeling these classes do not appear to affect the physics of the theory we investigate the possibility that these moduli could be only the consequence of a poor choice in the fine tuning of the mathematical setting we show that by simply choosing a minor extension of the functional class of the classical fields and coordinates the moduli disappear the knot classes become countable and the kinematical hilbert space of loop quantum gravity becomes separable

a complete classification of bifurcation diagrams for a class of p q laplace equations abstract we study the bifurcation problem of positive solutions for the one dimensional p q laplace equation with nonlinear term u r 1 there are five types of order relations for p q r we investigate the exact shape of the bifurcation curve in each type of the order relation we prove that there are two types of bifurcation curves that are increasing two types that are decreasing and one that is not monotone and turns exactly once moreover we study the asymptotic profile of the normalized solution u x u as u 0 or u where u denotes the l norm of u

a positive muscl hancock scheme for ideal magnetohydrodynamics we present a highly robust second order accurate scheme for the euler equations and the ideal mhd equations the scheme is of predictor corrector type with a muscl scheme following as a special case the crucial ingredients are an entropy stable approximate riemann solver and a new spatial reconstruction that ensures positivity of mass density and pressure for multidimensional mhd a new discrete form of the powell source terms is vital to ensure the stability properties the numerical examples show that the scheme has superior stability compared to standard schemes while maintaining accuracy in particular the method can handle very low values of pressure i e low plasma b or high mach numbers and low mass densities

projective loop quantum gravity i state space instead of formulating the state space of a quantum field theory over one big hilbert space it has been proposed by kijowski to describe quantum states as projective families of density matrices over a collection of smaller simpler hilbert spaces beside the physical motivations for this approach it could help designing a quantum state space holding the states we need in a latter work by okolow the description of a theory of abelian connections within this framework was developed an important insight being to use building blocks labeled by combinations of edges and surfaces the present work generalizes this construction to an arbitrary gauge group g in particular g is neither assumed to be abelian nor compact this involves refining the definition of the label set as well as deriving explicit formulas to relate the hilbert spaces attached to different labels if the gauge group happens to be compact we also have at our disposal the well established ashtekar lewandowski hilbert space which is

positive solutions to classes of infinite semipositone p q laplace problems with nonlinear boundary conditions abstract we consider one dimensional p q laplace problems u h t f u t 0 1 u 0 0 a u 1 g u 1 u 1 where 0 a 0 s s p 2 s s q 2 s 1 p q h c 0 1 0 f c 0 r with lim s 0 f s 0 and g c 0 0 0 such that g r s s is nondecreasing with respect to s 0 classifying the behaviors of f near infinity we establish the existence multiplicity and nonexistence of positive solutions in particular we provide a sufficient condition on f to obtain a multiplicity result for the case when lim s f s s r 1 0 1 r q which is new even in semilinear problems p q 2 the proofs are based on a krasnoselskii type fixed point theorem which is fit to overcome a lack of homogeneity

representations of the weyl algebra in quantum geometry the weyl algebra mathfrak a of continuous functions and exponentiated fluxes introduced by ashtekar lewandowski and others in quantum geometry is studied it is shown that in the piecewise analytic category every regular representation of mathfrak a having a cyclic and diffeomorphism invariant vector is already unitarily equivalent to the fundamental representation additional assumptions concern the dimension of the underlying analytic manifold at least three the finite wide triangulizability of surfaces in it to be used for the fluxes and the naturality of the action of diffeomorphisms but neither any domain properties of the represented weyl operators nor the requirement that the diffeomorphisms act by pull backs for this the general behaviour of c algebras generated by continuous functions and pull backs of homeomorphisms as well as the properties of stratified analytic diffeomorphisms are studied additionally the paper includes also a short and direct proof of the irreducibility of mathfrak a

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

high order well balanced finite volume schemes for simulating wave propagation in stratified magnetic atmospheres wave propagation in idealized stellar atmospheres is modeled by the equations of ideal mhd together with the gravity source term the waves are modeled as small perturbations of isothermal steady states of the system we consider a formulation of ideal mhd based on the godunov powell form with an embedded potential magnetic field appearing as a parameter the equations are discretized by finite volume schemes based on approximate riemann solvers of the hll type and upwind discretizations of the godunov powell source terms local hydrostatic reconstructions and suitable discretization of the gravity source term lead to a well balanced scheme i e a scheme which exactly preserves a discrete version of the relevant steady states higher order of accuracy is obtained by employing suitable minmod eno and weno reconstructions based on the equilibrium variables to construct a well balanced scheme the resulting high order well balanced schemes are validated on a suite of numerical experiments involving complex magnetic fields the schemes are observed to be robust and resolve the complex physics well

provably positive high order schemes for ideal magnetohydrodynamics analysis on general meshes this paper proposes and analyzes arbitrarily high order discontinuous galerkin dg and finite volume methods which provably preserve the positivity of density and pressure for the ideal magnetohydrodynamics mhd on general meshes unified auxiliary theories are built for rigorously analyzing the positivity preserving pp property of numerical mhd schemes with a harten lax van leer hll type flux on polytopal meshes in any space dimension the main challenges overcome here include establishing certain relation between the pp property and a discrete divergence of magnetic field on general meshes and estimating proper wave speeds in the hll flux to ensure the pp property in the 1d case we prove that the standard dg and finite volume methods with the proposed hll flux are pp under a condition accessible by a pp limiter for the multidimensional conservative mhd system the standard dg methods with a pp limiter are not pp in general due to the effect of unavoidable divergence error in the magnetic field we construct provably pp high order dg and finite volume schemes by proper discretization of the symmetrizable mhd system with two divergence controlling techniques the locally divergence free elements and suitably discretized godunov powell source term the former technique leads to zero divergence within each cell while the latter controls the divergence error across cell interfaces our analysis reveals in theory that a coupling of these two techniques is very important for positivity preservation as they exactly contribute the discrete divergence terms which are absent in standard multidimensional dg schemes but crucial for ensuring the pp property several numerical tests further confirm the pp property and the effectiveness of the proposed pp schemes unlike the conservative mhd system the exact smooth solutions of the symmetrizable mhd system are proved to retain the positivity even if the divergence free condition is not satisfied our analysis and findings further the understanding at both discrete and continuous levels of the relation between the pp property and the divergence free constraint

qcd on an infinite lattice we construct a mathematically well defined framework for the kinematics of hamiltonian qcd on an infinite lattice in mathbb r 3 and it is done in a c algebraic context this is based on the finite lattice model for hamiltonian qcd developed by kijowski rudolph e a to extend this model to an infinite lattice we need to take an infinite tensor product of nonunital c algebras which is a nonstandard situation we use a recent construction for such situations developed by grundling and neeb once the field c algebra is constructed for the fermions and gauge bosons we define local and global gauge transformations and identify the gauss law constraint the full field algebra is the crossed product of the previous one with the local gauge transformations the rest of the paper is concerned with enforcing the gauss law constraint to obtain the c algebra of quantum observables for this we use the method of enforcing quantum constraints developed by grundling and hurst in particular the natural inductive limit structure of the field algebra is a central component of the analysis and the constraint system defined by the gauss law constraint is a system of local constraints in the sense of grundling and lledo using the techniques developed in that area we solve the full constraint system by first solving the finite local systems and then combining the results appropriately we do not consider dynamics

quantum group connections the ahtekar isham c algebra known from loop quantum gravity is the algebra of continuous functions on the space of generalized connections with a compact structure lie group the algebra can be constructed by some inductive techniques from the c algebra of continuous functions on the group and a family of graphs embedded in the manifold underlying the connections we generalize the latter construction replacing the commutative c algebra of continuous functions on the group by a noncommutative c algebra defining a compact quantum group

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

extensions of a result by g talenti to p q laplace equations abstract we prove a comparison result for solutions to p q laplace equation via schwarz symmetrization for the p laplace equation the corresponding result is due to giorgio talenti in a special radial case we also prove a reverse comparison result

an entropy satisfying two speed relaxation system for the barotropic euler equations application to the numerical approximation of low mach number flows in the first part of this work we introduce a new relaxation system in order to approximate the solutions to the barotropic euler equations we show that the solutions to this two speed relaxation model can be understood as viscous approximations of the solutions to the barotropic euler equations under appropriate sub characteristic conditions our relaxation system is a generalization of the well known suliciu relaxation system and it is entropy satisfying a godunov type finite volume scheme based on the exact resolution of the riemann problem associated with the relaxation system is deduced as well as its stability properties in the second part of this work we show how the new relaxation approach can be successfully applied to the numerical approximation of low mach number flows we prove that the underlying scheme satisfies the well known asymptotic preserving property in the sense that it is uniformly first order accurate with respect to the mach number and at the same time it satisfies a fully discrete entropy inequality this discrete entropy inequality allows us to prove strong stability properties in the low mach regime at last numerical experiments are given to illustrate the behaviour of our scheme

an hllc type approximate riemann solver for ideal magnetohydrodynamics this paper presents a solver based on the hllc harten lax van leer contact wave approximate nonlinear riemann solver for gas dynamics for the ideal magnetohydrodynamics mhd equations written in conservation form it is shown how this solver also can be considered a modification of linde s adequate solver this approximation method is intended to resolve slow alfven and contact waves better than the original hll harten lax van leer solver compared to exact nonlinear solvers and roe s solver this new solver is computationally inexpensive in addition the method will exactly resolve isolated contacts and fast shocks the method also preserves positive density and pressure with two caveats first the numerical signal velocities the eigenvalues of the roe average matrix do not underestimate the physical signal velocities and second in a very few cases it may be required to change the wavespeeds of the riemann fan for the underlying hll method to guarantee positive pressures these conditi

projective loop quantum gravity ii searching for semi classical states in the first paper of this series an extension of the ashtekar lewandowski state space of loop quantum gravity was set up with the help of a projective formalism introduced by kijowski the motivation for this work was to achieve a more balanced treatment of the position and momentum variables also known as holonomies and fluxes while this is the first step toward the construction of states semi classical with respect to a full set of observables one uncovers a deeper issue which we analyse in the present article in the case of real valued holonomies specifically we show that in this case there does not exist any state on the holonomy flux algebra in which the variances of the holonomy and flux observables would all be finite let alone small it is important to note that this obstruction cannot be bypassed by further enlarging the quantum state space for it arises from the structure of the algebra itself a way out would be to suitably restrict the algebra of observables we take the first ste

uniqueness of the representation in homogeneous isotropic lqc we show that the standard representation of homogeneous isotropic loop quantum cosmology lqc is the gns representation that corresponds to the unique state on the reduced quantum holonomy flux algebra that is invariant under residual diffeomorphisms both when the standard algebra is used as well as when one uses the extended algebra proposed by fleischhack more precisely we find that in both situations the gns hilbert spaces coincide and that in the fleischhack case the additional algebra elements are just mapped to zero operators in order for the residual diffeomorphisms to have a well defined action on the quantum algebra we have let them act on the fiducial cell as well as on the dynamical variables thereby recovering covariance consistency with ashtekar and campilgia in the bianchi case is also shown

algebra of observables and charge superselection sectors for qed on the lattice quantum electrodynamics on a finite lattice is investigated within the hamiltonian approach first the structure of the algebra of lattice observables is analyzed and it is shown that the charge superselection rule holds next for every eigenvalue of the total charge operator a canonical irreducible representation is constructed and it is proved that all irreducible representations corresponding to a fixed value of the total charge are unique up to unitary equivalence the physical hilbert space is by definition the direct sum of these superselection sectors finally lattice quantum dynamics in the heisenberg picture is formulated and the relation of our approach to gauge fixing procedures is discussed

on the algebra of quantum observables for a certain gauge model we prove that the algebra of observables of a certain gauge model is generated by unbounded elements in the sense of woronowicz the generators are constructed from the classical generators of invariant polynomials by means of geometric quantization

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