conformal operators on forms and detour complexes on einstein manifolds for even dimensional conformal manifolds several new conformally invariant objects were found recently invariant differential complexes related to but distinct from the de rham complex these are elliptic in the case of riemannian signature the cohomology spaces of these conformally stable form spaces that we may view as spaces of conformal harmonics operators that generalise branson s q curvature global pairings between differential form bundles that descend to cohomology pairings here we show that these operators spaces and the theory underlying them simplify significantly on conformally einstein manifolds we give explicit formulae for all the operators concerned the null spaces for these the conformal harmonics and the cohomology spaces are expressed explicitly in terms of direct sums of subspaces of eigenspaces of the form laplacian for the case of non ricci flat spaces this applies in all signatures and without topological restrictions in the case of riemannian signature and compact manifolds this leads to new results on the global invariant pairings including for the integral of q curvature against the null space of the dimensional order conformal laplacian of graham et al

nonstandard analysis and jump conditions for converging shock waves nonstandard analysis is an area of modern mathematics that studies abstract number systems containing both infinitesimal and infinite numbers this article applies nonstandard analysis to derive jump conditions for one dimensional converging shock waves in a compressible inviscid perfect gas it is assumed that the shock thickness occurs on an infinitesimal interval and the jump functions in the thermodynamic and fluid dynamic parameters occur smoothly across this interval predistributions of the heaviside function and the dirac delta measure are introduced to model the flow parameters across a shock wave the equations of motion expressed in nonconservative form are then applied to derive unambiguous relationships between the jump functions for the flow parameters

on discrete fourth order boundary value problems with three parameters in this paper the existence multiplicity and nonexistence results of nontrivial solutions are obtained for discrete nonlinear fourth order boundary value problems with three parameters the methods used here are based on the critical point theory and monotone operator theory

path space measure for the 3 1 dimensional dirac equation in momentum space nonstandard analysis is used to construct a measure over paths for the path integral solution to the dirac equation in 3 1 dimension paths are considered in momentum space because the green function in the configuration space contains a derivative of function which keeps us from assigning a measure over paths the solution is obtained not only as the standard part of a nonstandard path sum with respect to a nonstandard measure but also as a standard path integral with respect to a standard measure extracted from the nonstandard one the result is an extension of gaveau s work b gaveau j funct anal 58 310 319 1984

on the existence of positive solutions of fourth order difference equations in this paper by means of krasnosel skii s fixed point theorem the existence of positive solutions of fourth order difference equations is considered

existence and multiplicity of solutions of a kind of fourth order boundary value problem abstract in this paper the existence and multiplicity results of the solutions are obtained for the fourth order boundary value problem u 4 t f t u t for all t 0 1 subject to u 0 u 1 u 0 u 1 0 where f is continuous the monotone operator theory and critical point theory are employed to discuss this problem respectively

multiplicity results for nonlinear periodic fourth order difference equations with parameter dependence and singularities abstract this paper is devoted to the study of nonlinear singular and non singular fourth order difference equations coupled with periodic boundary value conditions in the paper some existence and nonexistence results are given the results are deduced from krasnoselskii s fixed point theorems in cones an exhaustive study of the green s function related to a linear fourth order problem is done

the space of state vectors a hyperfinite approach we present a version of the formalism of dirac based on nonstandard analysis allowing us to deal with state vectors and operators using the resources of finite dimensional linear algebra the space of state vectors is a nonstandard hilbert space with hyperfinite dimension which includes all square integrable functions together with vectors representing states of definite position or momentum every vector is normalizable even when its norm is infinite observables are represented by hermitian operators which are always hyper bounded and defined on the whole space the connection with the standard theory is established by postulating the existence of hyper observables and nonstandard states each observable in the usual sense appears as a kind of standard scale approximation of some hyper observable we show that the probabilistic predictions are consistent with those of the standard theory consistency extends to time evolution in the sense that if an initial nonstandard state is near standard then the state after a finite time shall be infinitely near the standard state obtained through the schrodinger equation

multiple positive solutions of singular and nonsingular discrete problems via variational methods we employ the critical point theory to establish the existence of multiple solutions of some regular as well as singular discrete boundary value problems 2004 elsevier ltd all rights reserved msc 34b16 39a10

existence of positive solutions for bvps of fourth order difference equations this paper is concerned with one type of boundary value problems bvps sufficient conditions for the existence of positive solutions of these bvps are derived and for a special case there is an upper bound of the set of l for the existence

positive solutions of fourth order boundary value problems with two parameters abstract in this paper the existence results of positive solutions are obtained for fourth order boundary value problem u 4 u u f t u 0 where f 0 1 r r is continuous r and satisfy

existence multiplicity and infinite solvability of positive solutions to a nonlinear fourth order periodic boundary value problem abstract the existence of n and infinitely many positive solutions is proved for the nonlinear fourth order periodic boundary value problem u 4 t u t u t f t u t 0 t 1 u i 0 u i 1 i 0 1 2 3 where n is an arbitrary natural number and 2 2 0 1 2 2 2 2 4 2 1 0 this kind of fourth order boundary value problems usually describes the equilibrium state of an elastic beam with periodic boundary condition the main results show that the problem may have n or infinitely many positive solutions provided the growth rates of nonlinear term f t l are appropriate on some bounded subsets of its domain

transformation techniques and fixed point theories to establish the positive solutions of second order impulsive differential equations abstract using a new method for dealing with the impulse term of second impulsive differential equations we are concerned with determining values of for which there exist positive solutions of the following boundary value problems in the form of x t f t x t 0 t j t t k x t k x t k c k x t k k 1 2 n a x 0 b x 0 a x 1 b x 1 0 1 h s x t d t where 0 is a positive parameter c k is a real sequence with c k 1 k 1 2 n the dependence of positive solution x t on the parameter is also studied i e lim x or lim x 0 the proof of our main result is based upon transformation techniques and fixed point theories this is probably the first time the existence of positive solutions of second order impulsive differential equations has been studied by applying this technique

existence of solutions for some fourth order boundary value problems with parameters abstract in this paper existence and multiplicity results for solutions are obtained for the fourth order boundary value problem bvp u 4 t u t u t f t u t 0 t 1 u 0 u 1 u 0 u 1 0 where f 0 1 r r is continuous r and r are parameters by using the critical point theory and morse theory we obtain that if satisfy 4 2 1 then the above bvp has solutions where is in some different intervals

three positive solutions of boundary value problems for p laplacian difference equations in this paper existence criteria for three positive solutions of p laplacian difference equation f p u t 1 a t f u t 0 t 1 t 1 u 0 u t 2 0 oru 0 u t 1 0 are established by using the five functionals fixed point theorem as an application one example is given to illustrate the main result in this paper

existence and multiplicity of solutions for fourth order boundary value problems with parameters abstract in this paper we study the existence and multiplicity of the solutions for the fourth order boundary value problem bvp u 4 t u t u t f t u t 0 t 1 u 0 u 1 u 0 u 1 0 where f 0 1 r r is continuous r and r are parameters by means of the idea of the decomposition of operators shown by chen w y chen a decomposition problem for operators xuebao of dongbei renmin university 1 1957 95 98 see also m krasnosel skii topological methods in the theory of nonlinear integral equations gostehizdat moscow 1956 and the critical point theory we obtain that if the pair is on the curve 2 4 satisfying 2 2 then the above bvp has at least one two three and infinitely many solutions for being in different interval respectively

existence and uniqueness of positive periodic solutions of functional differential equations abstract in this paper we consider the existence and uniqueness of positive periodic solution for the periodic equation y t a t y t h t f y t t by the eigenvalue problems of completely continuous operators and theory of concave or convex operators and its eigenvalue we establish some criteria for existence and uniqueness of positive periodic solution of above functional differential equations with parameter in particular the unique solution y t of the above equation depends continuously on the parameter finally as an application we obtain sufficient condition for the existence of positive periodic solutions of the nicholson blowflies model

positive solutions of a general discrete boundary value problem in this paper the existence of positive solutions for a nonlinear general discrete boundary value problem is established such results extend and improve some known facts for the two point and three point boundary value problems particularly the boundary value conditions can be nonlinear and the method is new for explaining the main results some numerical examples are also given

three solutions for a p laplacian boundary value problem with impulsive effects in this paper a p laplacian boundary value problem with impulsive effects is considered multiplicity of solutions is obtained by three critical points theorem an example is presented to illustrate main result

density functional approach to quantum lattice systems for quantum lattice systems we consider the problem of characterizing the set of single particle densities which come from the ground state eigenspace of somen particle hamiltonian of the form h 0 sum nolimits i 1 n v x i whereh0 is a fixed bounded operator representing the kinetic and interaction energies we show that the conditions on are that it be strictly positive properly normalized and consistent with the pauli principle our results are valid for both finite and infinite lattices and for either bosons or fermions the coulomb interaction may be included inh0 if the lattice dimension is 2 we also characterize those single particle densities which come from the gibbs states of such hamiltonians at finite temperature in addition to the conditions stated above must satisfy a finite entropy condition

existence multiplicity and dependence on a parameter for a periodic boundary value problem the authors consider the boundary value problem y 2 y g t f y 0 0 t 2 y 0 y 2 y 0 y 2

coarse grained spin density functional theory infinite volume limit via the hyperfinite coarse grained spin density functional theory sdft is a version of sdft which works with number spin densities specified to a limited resolution averages over cells of a regular spatial partition and external potentials constant on the cells this coarse grained setting facilitates a rigorous investigation of the mathematical foundations which goes well beyond what is currently possible in the conventional formulation problems of existence uniqueness and regularity of representing potentials in the coarse grained sdft setting are here studied using techniques of robinsonian nonstandard analysis every density which is nowhere spin saturated is v representable and the set of representing potentials is the functional derivative in an appropriate generalized sense of the lieb internal energy functional quasi continuity and closure properties of the set valued representing potentials map are also established the extent of possible non uniqueness is similar to that found in non rigorous studies

the principal eigenvalue of a conformally invariant differential operator with an application to semilinear elliptic pde in this paper we give conditions under which the paneitz operator a conformally invariant fourth order differential operator is positive we also prove a sharp estimate for a related conformal invariant these results are combined to study the existence of a semilinear pde with critical sobolev growth

pointwise analysis of scalar fields a nonstandard approach a new nonstandard analytical approach to quantum fields is presented which gives a mathematical foundation for manipulating pointwise defined quantum fields in our approach a field operator x is not a standard operator valued distribution but a nonstandard operator valued function then formal expressions containing e g x 2 can be understood literally and shown to be well defined in the free field cases we show that the wightman functions are explicitly calculated with the pointwise field without any regularization e g wick product our notion of pointwise fields is applied also to the path integral formalisms of scalar fields we show that some of physicists naive expressions of lagrangian path integral formulas can be rigorously justified

hohenberg kohn theorem and non v representable densities abstract in the density functional formalism of hohenberg and kohn1 the variation is only allowed over the one particle densities which are pure state v representable ps v representable levy2 and lieb3 proved that not every ensemble v representable e v representable density is ps v representable since we show that the hohenberg kohn formalism can be extended to a variation over e v representable densities for degenerated ground states levy s and lieb s result is not a counterexample to the universality of the hohenberg kohn theorem the question whether every n representable density is e v representable has remained open so far presenting examples of non e v representable densities we answer this question in the negative thus the value of levy s functional4 for the calculation of ground state energies is obvious since this functional only requires the n representability of the densities therefore we transfer two approaches for the calculation of excited state energies into the framework of levy s formalism

discrete fourth order boundary value problems with four parameters abstract the theory of nonlinear difference equations and discrete boundary value problems has been widely used to study discrete models in many fields such as computer science economics mechanical engineering control systems artificial or biological neural networks ecology cybernetics and so on fourth order difference equations derived from various discrete elastic beam problems in this paper we seek further study of the multiplicity results for discrete fourth order boundary value problems with four parameters in fact using a consequence of the local minimum theorem due bonanno we look for the existence one solution under algebraic conditions on the nonlinear term and two solutions for the problem under algebraic conditions with the classical ambrosetti rabinowitz ar condition on the nonlinear term furthermore by employing two critical point theorems one due averna and bonanno and another one due bonanno we guarantee the existence of two and three solutions for our problem in a special case

multiple solutions of some nonlinear fourth order beam equations abstract several new existence theorems on three solutions and infinitely many solutions for the following fourth order beam equation are obtained u 4 f t u t t 0 1 u 0 u 1 u 0 u 1 0 where f c 1 0 1 r 1 r 1 the morse theory is employed to discuss this problem

on some discrete boundary value problem with parameters abstract we consider the existence and multiplicity of solutions to discrete boundary value problems with the use of critical point theory combined with some monotonicity argument we consider both sub and super linear problems

twistor theory at fifty from contour integrals to twistor strings we review aspects of twistor theory its aims and achievements spanning the last five decades in the twistor approach space time is secondary with events being derived objects that correspond to compact holomorphic curves in a complex threefold the twistor space after giving an elementary construction of this space we demonstrate how solutions to linear and nonlinear equations of mathematical physics anti self duality equations on yang mills or conformal curvature can be encoded into twistor cohomology these twistor correspondences yield explicit examples of yang mills and gravitational instantons which we review they also underlie the twistor approach to integrability the solitonic systems arise as symmetry reductions of anti self dual asd yang mills equations and einstein weyl dispersionless systems are reductions of asd conformal equations we then review the holomorphic string theories in twistor and ambitwistor spaces and explain how these theories give rise to remarkable new formulae for the computation of quantum scattering amplitudes finally we discuss the newtonian limit of twistor theory and its possible role in penrose s proposal for a role of gravity in quantum collapse of a wave function

three symmetric solutions of lidstone boundary value problems for difference and partial difference equations we consider the boundary value problem 2my k m f y k 2y k 1 2iy k 1 2 m 1 y k m 1 k a 1 b 1 2iy a 1 m 2iy b 1 m 2i 0 0 i m 1 where m 1 and 1 m f rm 0 is continuous by using amann and leggett williams fixed point theorems we develop growth conditions on f so that the boundary value problem has triple positive symmetric solutions the results obtained are then applied in the investigation of radial solutions for certain partial difference equation subject to lidstone type conditions

existence of a positive solution for one dimensional singular p laplacian problems and its parameter dependence article history we consider one dimensional singular p laplacian problems of the form q 1 and may be singular at t 0a nd ort 1 using fixed point techniques combined with the partially ordered structure of banach space we establish some new and more general existence multiplicity and nonexistence results we also study the dependence of the positive solution u t on the parameter t hat is

existence of solutions for a nonlinear algebraic system with a parameter abstract taking advantage of a recent critical point theorem the existence of infinitely many solutions for a nonlinear algebraic system with a parameter is established our goal was achieved requiring an appropriate behavior of the nonlinear term f either at zero or at infinity without symmetry conditions in addition for a suitable class of systems a strong discrete maximum principle is presented