on spacelike hypersurfaces with constant mean curvature in the de sitter space soit n n n 1 c g une variete de lorentz a n 1 dimensions de courbure constante positive c soit m une hypersurface du genre espace complete a courbure moyenne constante h dans n on suppose que 1 h c 1 2 quand n 2 2 h 2 n 1 c 1 2 n quand n 3 alors m est totalement ombilicale

oscillatory and asymptotic behavior of certain functional differential equations some new criteria for the oscillatory and asymptotic behavior of odd order functional equations of the form x n t p t x n 1 t q t f x g t are established

on the oscillation of certain functional differential equations via comparison methods abstract several new criteria for the oscillation of the functional differential equations of the form d dt 1 a n 1 t d dt 1 a n 2 t d dt 1 a 1 t d dt x t q t f x g t 0 are established in this paper

oscillation theorems for certain functional differential equations abstract some new criteria for the oscillation of functional differential equations with a middle term of the form x n t p t x n 1 t h h t x g t are presented

oscillatory and asymptotic behavior of delay differential equations with a nonlinear damping term in this paper we study the oscillatory and asymptotic behavior of the solutions of delay differential equations of the form x n t f t x t x t 0 n even by comparing with certain differential equations of the same or lower order whose behavior is known the obtained criteria can be applied to the delay differential equation x n t q t x t x t sign x t 0 n even where and are positive constants

kamenev type oscillation criteria for even order neutral differential equations with deviating arguments abstract sufficient conditions are established for the oscillation of even order neutral differential equations of the form x t i 1 m p i t x i t n j 1 l q j t f j x j t 0 t t 0 where n 2 is even integer

vertex operators for twisted quantum affine algebras we construct explicitly the q vertex operators intertwining operators for the level one modules v i of the classical quantum affine algebras of twisted types using interacting bosons where i 0 1 for a 2 n 1 i 0 for d 3 i 0 n for d 2 1 and i n for a 2 n a perfect crystal graph for d 3 4 is constructed as a by product

oscillation theorems for second order nonlinear functional differential equations with damping abstract new oscillation criteria are given for a second order nonlinear functional differential equation t t p t t q t g t sgn g t 0 where 1 and 0

space like submanifolds in de sitter spaces let m n be a space like submanifold in a de sitter space mpn p c with flat normal bundle this paper gives some intrinsic conditions for m n to be totally umbilical

integral inequalities and second order linear oscillation on the interval 0 a u 0 below by a solution we mean a non trivial solution a solution is oscillatory if it has an infinite number of zeros numerous oscillation criteria are known and diversified approaches have proved fruitful among these is the use of the associated riccati equation although the theory of differential and integral inequalities has been used by some investigators see e g hartman 4 71 the authors believe that it has not been fully exploited the non oscillation of 1 1 is equivalent to the existence of a solution of the associated riccati equation on a half line thus one technique of proving oscillation is to show that any solution of the riccati equation diverges to i co at a finite point the theory of differential and integral inequalities comes in handy in the estimation of the growth of such a solution below we describe some known results that are relevant to results established in this paper although these are stated here for the case i co in the rest of the paper we consider both cases a 0 and a 00 it is well known that if q is large in the mean then 1 1 is oscillatory a typical example of such a criterion is due to fite leighton and wintner if

on some properties of solution of second order nonlinear differential equations on etudie le comportement oscillatoire des solutions de l equation differentielle non lineaire d ordre 2 sur t 0 a t x x q t f x t

nonoscillation for second order sublinear dynamic equations on time scales consider the emden fowler sublinear dynamic equation 0 1 x d d t p t f x s t 0 where p c t r t is a time scale f x i 1 ma ix b i where a i 0 0 0 c a and the sublinear q difference equation x d d t b 1 nt cx a qt 0 0 1 b 0 c 1 a

rotation hypersurfaces in spaces of constant curvature rotation hypersurfaces in spaces of constant curvature are defined and their principal curvatures are computed a local characterization of such hypersurfaces with dimensions greater than two is given in terms of principal curvatures some special cases of rotation hypersurfaces with constant mean curvature in hyperbolic space are studied in particular it is shown that the well known conjugation between the helicoid and the catenoid in euclidean three space extends naturally to hyperbolic three space h3 in the latter case catenoids are of three different types and the explicit correspondence is given it is also shown that there exists a family of simply connected complete embedded nontotally geodesic stable minimal surfaces in h3

oscillation criteria for certain nth order differential equations with deviating arguments abstract oscillation criteria for nth order differential equations with deviating arguments of the form x n 1 t 1 x n 1 t f t x g t 0 n even are established where g c t0 r f c t0 r r and 0 is a constant

oscillations caused by several retarded and advanced arguments in this paper we study the oscillatory behavior of equations of the forms y t i 1npiy t i 0 and y t i 1npiy t i 0 where piand i i 1 2 n are positive constants we prove that each one of the following conditions 1 pi i1 e for some i i 1 2 n 2 i 1npi 1 e where min 1 2 n 3 i 1npi 1 n i 1n i 1 e or 4 1 n i 1n pi i 1 2 21 e implies that every solution of and oscillates a generalization in the case where the coefficientspi i 1 2 n are positive and continuous functions is also presented

linearization of second order nonlinear oscillation theorems the problem of oscillation of super and sublinear emden fowler equations is studied established are a number of oscillation theorems involving comparison with related linear equations recent results on linear oscillation can thus be used to obtain interesting oscillation criteria for nonlinear equations 25 references

new characterizations for hyperbolic cylinders in anti de sitter spaces abstract in this paper we study complete maximal spacelike hypersurfaces in anti de sitter space h 1 n 1 with either constant scalar curvature or constant non zero gauss kronecker curvature we characterize the hyperbolic cylinders h m c 1 h n m c 2 1 m n 1 as the only such hypersurfaces with n 1 principal curvatures with the same sign everywhere in particular we prove that a complete maximal spacelike hypersurface in h 1 5 with negative constant gauss kronecker curvature is isometric to h 1 c 1 h 3 c 2

oscillation theorems for nth order differential equations with deviating arguments abstract this paper is concerned with the study of the oscillatory behavior of solutions of the n th order nonlinear differential equation a t x n v t v q t f x g t 0 where n is even and 1 v n 1 a systematic study is attempted which extends and correlates a number of existing results

on the oscillation of certain forced functional differential equations abstract some new criteria for the oscillation of a certain class of forced functional differential equations of the form formula are established here we require the function f to be locally of bounded variation

comparison theorems for forced functional differential equations abstract in this paper we establish comparison results by which oscillating and asymptotic properties of solutions of forced differential equations with diviating arguments of the form d dt 1 a n 1 t d dt d dt 1 a 1 t d dt x t h t x g t q t are inherited from the same properties of solutions of the unforced ordinary equation d dt 1 a n 1 t d dt d dt 1 a 1 t d dt x t h t x t 0 the obtained results extend the oscillation criteria of trench and kusano and naito to the more general equation

an application of integral inequality to second order nonlinear oscillation abstract a simple result concerning integral inequalities enables us to give an alternative proof of waltman s theorem lim t t 0 a s ds implies oscillation of the second order nonlinear equation y t a t f y t 0 to prove an analog of wintner s theorem that relates the nonoscillation of the second order nonlinear equations to the existence of solutions of some integral equations assuming that lim t t 0 a s ds exists and to give an alternative proof and to extend a result of butler an often used condition on the coefficient a t is given a more familiar equivalent form and an oscillation criterion involving this condition is established

strong and weak oscillatory behavior in time varying linear difference equations of arbitrary order in the presence of unmodeled dynamics and nominal parametrical errors abstract this paper is devoted to investigate the presence of oscillating solutions in time varying difference equations of arbitrary n th order even in the case when there are parametrical errors i e errors in the sequences defining their coefficients or unmodeled dynamics i e the current order n is unknown with n n 0 the nominal known order the problem is formulated linked to the concepts of conjugacy disconjugacy positivity and generalized zeros and very general conditions of oscillation are obtained both over particular intervals and for the whole solution some results concerned with the presence of stable asymptotically stable oscillatory solutions are also presented

new characterizations of hyperbolic cylinders in semi riemannian space forms abstract through the study of the so called linear weingarten spacelike submanifolds immersed with parallel normalized mean curvature vector in a semi riemannian space form q p n p c of index p and constant sectional curvature c 1 0 1 our aim in this paper is establish new characterizations for the hyperbolic cylinders of q p n p c our approach is based on the use of a simons type formula related to an appropriated cheng yau modified operator jointly with the application of suitable generalized maximum principles

oscillatory properties of functional differential equations abstract in the present paper we establish some new criteria for the oscillatory and asymptotic behavior of the functional differential equations with deviating arguments of the form d dt 1 a n 1 t d dt 1 a n 2 t d dt 1 a 1 t d dt x t t x g 1 t x g m t 0 where 1 the obtained results improve unify and correlate a number of existing criteria

the riemannian product structures of spacelike hypersurfaces with constant k th mean curvature in the de sitter spaces abstract in this paper we investigate complete spacelike hypersurfaces in the de sitter space s 1 n 1 c with constant k th mean curvature and two distinct principal curvatures one of which is simple we obtain some characterizations of the riemannian product h 1 c 1 s n 1 c 2 or h n 1 c 1 s 1 c 2 in the de sitter space s 1 n 1 c

on the complete linear weingarten spacelike hypersurfaces with two distinct principal curvatures in lorentzian space forms abstract we deal with complete linear weingarten spacelike hypersurfaces immersed in a lorentzian space form having two distinct principal curvatures in this setting we show that such a spacelike hypersurface must be isometric to a certain isoparametric hypersurface of the ambient space under suitable restrictions on the values of the mean curvature and of the norm of the traceless part of its second fundamental form our approach is based on the use of a simons type formula related to an appropriated cheng yau modified operator jointly with some generalized maximum principles

oscillation criteria of a certain class of third order nonlinear delay differential equations with damping abstract in this paper we are concerned with the oscillation of third order nonlinear delay differential equations of the form r 2 t r 1 t y p t y q t f y g t 0 by using a generalized riccati transformation and integral averaging technique we establish some new sufficient conditions which insure that any solution of this equation oscillates or converges to zero in particular several examples are given to illustrate the importance of our results

oscillatory behaviour of higher order neutral type nonlinear forced differential equation with oscillating coefficients abstract oscillation criteria are given for higher order neutral type nonlinear forced differential equation with oscillating coefficients of the form y t p t y t n i 1 m q i t f i y i t s t

uniqueness of entire graphs in warped products abstract under suitable restrictions on the higher order mean curvatures we establish uniqueness results for entire graphs in a warped product space satisfying a standard convergence condition consequences are given for such graphs to be slices by using comparison inequalities of their higher order mean curvatures applications to the study of the rigidity of minimal and radial graphs in the euclidean space are also given

on the umbilicity of complete constant mean curvature spacelike hypersurfaces in this paper we show that a complete spacelike hypersurface immersed with constant mean curvature either in the de sitter space or in the anti de sitter space must be totally umbilical provided that its gauss mapping has some suitable behavior in particular we use an extension of hopf s maximum principle due to yau in order to give new positive answers to goddard s conjecture

oscillation of certain functional differential equations sufficient conditions are presented for all bounded solutions of the linear system of delay differential equations 1 m 1 frac d m y i t dt m sum limits j 1 n q ij y j t h jj 0 text m geqslant 1 i 1 2 n to be oscillatory where qije hjj 0 i j 1 2 n also we study the oscillatory behavior of all bounded solutions of the linear system of neutral differential equations 1 m 1 frac d m dt m y i t cy i t g sum limits j 1 n q ij y j t h 0 where c g and h are real constants and i 1 2 n

oscillation theorems for nth order delay differential equations abstract oscillation criteria are obtained for a class of functional differential equations including the generalized emden fowler equation x n q t x g t sgn x g t 0 0 as a special case the purpose of this paper is to extend and improve known fundamental oscillation criteria for superlinear sublinear and linear differential equations with retarded arguments

oscillation criteria for second order nonlinear differential equations with nonlinear damping this paper is concerned with the oscillation of a class of general type second order differential equations with nonlinear damping several new oscillation criteria are established under quite general assumptions which improve and extend the known results in the literature

oscillation of even order linear functional differential equations with deviating arguments of mixed type where n is even p 0 co r and g o 00 r are continuous p t 0 g t is nondecreasing and cm g t 03 by a proper solution of eq 1 we mean a function x tx 00 r which satisfies 1 for all sufficiently large t and sup lx t j t t 0 for any t tx we make the standing hypothesis that 1 does possess proper solutions a proper solution is called oscillatory if it has arbitrarily large zeros otherwise it is called nonoscillatory if x t is a nonoscillatory solution of l then there exist an even integer 1 e 0 2 n and a t tx such that

the oscillation of higher order differential equations with deviating arguments in this paper some new criteria for the oscillation of higher order functional differential equations of the form lnx t f t x g t 0 n is even are established some of our results are obtained via comparing it with second order ordinary linear and first order delay differential equations the oscillation of this equation when n is odd is also considered then we shall use the obtained results to study the oscillatory behavior of the neutral functional differential equations of the form ln x t c t x t f t x g t 0 n is even and the damped functional differential equations of the type lnx t h t x g t ddtx h t 0 n is even the obtained results extend improve and correlate a number of existing criteria

oscillatory and asymptotic behavior of solutions of differential equations with deviating arguments the oscillatory and asymptotic behavior of solutions of a class of nth order nonlinear differential equations with deviating arguments of the form e lnx t q t f x g1 t x gm t 0 where 1 and l0x t x t lkx t ak t lk 1x t k 1 2 n ddt is examined a classification of solutions of e with respect to their behavior as t and their oscillatory character is obtained the comparisons of e 1 and e 1 with first and second order equations of the form y t c1 t f y g1 t y gm t 0 and an 1 t z t c2 t f z g1 t z gm t 0 respectively are presented the obtained results unify extend and improve some of the results by graef grammatikopoulos and spikes philos and staikos

new calabi bernstein type results in lorentzian product spaces with density abstract we investigate the rigidity of complete spacelike hypersurfaces immersed in a weighted lorentzian product space of the type r 1 p f n endowed with a weight function f which does not depend on the parameter t r and whose riemannian fiber p n has nonnegative bakry emery ricci tensor in this direction supposing that the f mean curvature is constant and assuming appropriate constraints on the norm of the gradient of f we prove that such a spacelike hypersurface must be a slice of the ambient space as application of our investigation we obtain new calabi bernstein type results concerning entire spacelike graphs constructed over p n our approach is based on an extension for the drift laplacian of a generalized maximum principle of akutagawa 1987 and a liouville type result due to pigola et al 2005

asymptotic and oscillatory behavior of nth order forced functional differential equations abstract some new oscillation criteria for the forced functional differential equation x n t a t x q t e t are established

bosonic realizations ofuq c 1 n abstract we construct explicitly the quantum symplectic affine algebrauq sp 2n using bosonic fields the fock space decomposes into irreducible modules of level 1 2 quantizing the feingold frenkel construction forq 1

on certain riccati integral equations and second order linear oscillation are closely related is well known many important results in the oscillation theory of 1 1 are in fact established by studying either 1 2 or 1 3 recent studies suggest that a great deal can still be said about 1 2 and l 3 thus leading to new results on 1 1 see 7 91 particularly useful in these studies is the theory of differential and integral inequalities the present work further supports this view point in this paper we first give a comparison theorem section 2 theorem 1 for riccati integral equations of the form 1 3 involving monotonic rearrangements of the function q t ji q s ds as a corollary we establish an extension of the well known oscillation criteria of hille s 6 to more general potentials section 3 theorem 4 in section 4 we elaborate on a method of hartman to derive a comparison type result involving the square of the integral j h q s ds and then deduce a lyapunov type inequality for disfocality the preliminaries will be introduced in the appropriate sections 315 0022 247x 82 020315 16 02 00 o

on spacelike hypersurfaces with constant scalar curvature in locally symmetric lorentz spaces the purpose of this paper is to study compact or complete spacelike hypersurfaces with constant normalized scalar curvature in a locally symmetric lorentz space satisfying some curvature conditions we give an optimal estimate of the squared norm of the second fundamental form of such hypersurfaces furthermore the totally umbilical hypersurfaces are characterized

higher level representations of the quantum affine algebrauq ŝl 2 abstract we introduce the notion of quantum z algebras for the quantum affine algebra after developing the quantum z algebra we use it to study the quantum affine algebrauq sl 2 and construct bases for each of its higher level modules

integral criteria for oscillation of third order nonlinear differential equations abstract in this paper we are concerned with the oscillation of third order nonlinear differential equations of the form r 2 t r 1 t y p t y q t f y 0 t t 0 where t 0 0 is a fixed real number r 1 r 2 p q i t 0 r are nonnegative continuous functions with r 1 0 r 2 0 f r r is a continuous function by making use of a generalized riccati transformation we establish some new sufficient integral conditions under which the equation has at least one oscillatory solution examples are given to illustrate the importance of our results