the oscillation and asymptotically monotone solutions of second order quasilinear differential equations abstract we offer sufficient conditions for the oscillation of all solutions of the perturbed quasilinear differential equation a t y 1 y q t y p t y y as well as for the existence of a positive monotone solution of the damped differential equation a t y 1 y b t y 1 y h t y 0 where 0 examples that dwell upon the importance of our results are also included

log canonical thresholds of certain fano hypersurfaces we study log canonical thresholds on quartic threefolds quintic fourfolds and double spaces as an important application we show that they have kahler einstein metrics if they are general

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

oscillation and nonoscillation theorems for second order nonlinear difference equations abstract some new oscillation and nonoscillation theorems are obtained for second order nonlinear difference equations a p n a x n q n a x n f n x n b x n which generalize and improve all results proved by popenda in j math anal appl 123 1987 34 48

oscillation theorems for higher order nonlinear delay difference equations abstract some new oscillation results for difference equations of the form 1 1 and forced difference equations of the form 1 2 are established the results obtained are discrete analogues of some known results for delay differential equations

oscillations in a class of difference equations a class of linear difference equations is considered and a necessary and sufficient condition for the oscillation of solutions is established this condition is referred to the roots of the associated characteristic equation

oscillation of neutral difference equations with periodic coefficients abstract two classes of neutral linear difference equations with periodic coefficients are considered and necessary conditions and also sufficient conditions for the oscillation of all solutions are established these conditions are referred to the roots of the associated characteristic equations

on the oscillation of some linear difference equations with periodic coefficients abstract some linear difference equations with periodic coefficients not necessarily nonnegative are considered necessary conditions and sufficient conditions for the oscillation of the solutions are established conditions under which all nonoscillatory solutions tend to zero at are also presented the results obtained are the discrete analogues of the oscillation results for some linear delay differential equations with periodic coefficients which were given earlier by the second author oscillations of some delay differential equations with periodic coefficients j math anal appl 162 1991 452 475

fano varieties with many selfmaps abstract we study global log canonical thresholds of anticanonically embedded quasismooth weighted fano threefold hypersurfaces having terminal quotient singularities to prove the existence of a kahler einstein metric on most of them and to produce examples of fano varieties with infinite discrete groups of birational automorphisms

necessary and sufficient conditions for oscillations of higher order delay differential equations we obtain necessary and sufficient conditions under which ail solutions of certain n th order delay differential equations oscillate when n is odd and necessary and sufficient conditions under which all bounded solutions oscillate when n is even the conditions derived involve the characteristic equation of the delay differential equation

oscillations of difference equations with non monotone retarded arguments consider the first order retarded difference equation x n p n x n 0 n n 0 where p n n 0 is a sequence of nonnegative real numbers and n n 0 is a sequence of integers such that n n 1 n 0 and lim n n under the assumption that the retarded argument is non monotone a new oscillation criterion involving lim inf is established an example illustrates the case when the result of the paper implies oscillation while previously known results fail

on existence of positive solutions and bounded oscillations for neutral difference equations abstract a criterion for the existence of a positive solution for the first order difference equation xn cxn m pnxn k 0 pn 0 is established results are also obtained for the oscillation and nonoscillation of solutions of a second order difference equation 2 xn cxn m pnxn k where the pn are either of constant sign or are oscillatory some of the results are sharp

bounded solutions and wavefronts for discrete dynamics this paper deals with the second order nonlinear difference equation dd r k dd u k q kg u k 1 0 where r k and q k are positive real sequences defined on n cup 0 and the nonlinearity g r to r is nonnegative and nontrivial sufficient and necessary conditions are given for the existence of bounded solutions starting from a fixed initial condition u 0 the same dynamic with f instead of g such that uf u 0 for u not 0 was recently extensively investigated on the contrary our nonlinearity g is of a small appearance in the discrete case its introduction is motivated by the analysis of wavefront profiles in biological and chemical models the paper emphasizes the many different dynamical behaviors caused by such a g with respect to the equation involving function f some applications in the study of wavefronts complete this work

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

oscillation theorems for forced neutral difference equations abstract some oscillation criteria are established for forced neutral difference equations of the form x n px n k q n x n f n and 2 x n px n k q n x n f n the results remain valid even when f n 0

analysis of switched diversity systems on generalized fading channels the performances of switched diversity systems operating on generalized nakagami fading channels are analyzed using a discrete time model the average bit error rate ber of binary noncoherent frequency shift keying ncfsk on slow nonselective nakagami fading channels is derived closed form expressions that can be used to determine optimum switching thresholds in a minimum error rate sense are also derived in addition the use of optimum fixed thresholds is considered it is found that a considerable amount of diversity gain can be obtained using an optimum fixed rather than adaptive switching threshold results are obtained for both independent and correlated nakagami fading branch signals the effects of fading severity and the correlation coefficient on both the ber and on the optimum switching threshold are investigated it is shown that useful diversity gain can be obtained with power correlation coefficients as high as 0 9 when the fading is strong the results for a rayleigh channel are obtained and presented as a special case of generalized fading model

oscillation criteria for first order nonlinear delay difference equations in this paper two interesting oscillation criteria are obtained for all solutions of the nonlinear delay difference equation of the formx n 1 x n p nf x n l 1 x n l 2 x n l m 0 n 0 1 2 the results extend some well known results in the literature and two examples are given to demonstrate the advantage of our results

unified analysis of switched diversity systems in independent and correlated fading channels the moment generating function mgf of the signal power at the output of dual branch switch and stay selection diversity ssd combiners is derived the first order derivative of the mgf with respect to the switching threshold is also derived these expressions are obtained for the general case of correlated fading and nonidentical diversity branches and hold for any common fading distributions e g rayleigh nakagami m rician nakagami q the mgf yields the performance bit or symbol error probability of a broad class of coherent differentially coherent and noncoherent digital modulation formats with ssd reception the optimum switching threshold in a minimum error rate sense is obtained by solving a nonlinear equation which is formed by using the first order derivative of the mgf this nonlinear equation can be simplified for several special cases for independent and identically distributed diversity branches the optimal switching threshold in closed form is derived for three generic forms of the conditional error probability for correlated rayleigh or nakagami m fading with identical branches the optimal switching threshold in closed form is derived for the noncoherent binary modulation formats we show previously published results as special cases of our unified expression selected numerical examples are presented and discussed

boundedness and asymptotic behavior results for nonlinear difference equations with positive and negative coefficients this paper is concerned with the nonlinear delay difference equations with positive and negative coefficients x n 1 x n p n f x n k q n f x n l 0 n n 0 sufficient conditions are obtained under which every solution of eq is bounded and tends to a constant as n

oscillation of some nonlinear difference equations we investigate the oscillatory behavior of all solutions of a new class of first order nonlinear neutral difference equations several explicit oscillation criteria are established our main results are supported by illustrative examples

oscillation and comparison results for delay difference equations abstract we establish a necessary and sufficient condition for the existence of positive solutions and obtain sturm type comparison theorem for the difference equation x n 1 x n i 1 m pi n x n ki o n 0 we also obtain the following comparison result equation oscillates if and only if x n 1 x n i 1 m qi n x n ki o n 0 oscillates provided n lim pi n qi i 1 2 m

some new results for bounded and monotone properties of solutions of second order quasi linear forced difference equations abstract some new results are obtained for the bounded and monotone properties of solutions of second order quasi linear forced difference equations p n 1 y n 1 q n y n r n n n 0 where n 0 n 0 1 2 3 p n n n 0 1 is a positive sequence q n n n 0 is a nonnegative real sequence with q n 0 r n n n 0 is a real sequence and and are quotients of odd positive integers some errors in 1 are pointed out and addressed examples are given to illustrate the advantages of the new results

new oscillation criteria for delay difference equations abstract this paper is concerned with the oscillation of all solutions of the delay difference equations and where pn and pi n are sequences of nonnegative real numbers an k and ki are positive integers new oscillation criteria of the forms for equation and for equation are obtained where lim infn 1 k n 1i n k pi and c is the the best possible function of in some sense

nonexistence of positive solution of a class of nonlinear difference equation sufficient conditions are derived for the nonexistence of positive solution of a class of nonlinear difference equations the conditions are based on nonexistence of positive characteristic roots of the characteristic equation of the nonlinear difference equation

asymptotic and oscillatory behavior of solutions of certain second order nonlinear difference equations mingshu peng department of applied mathematics beijing institute of technology beijing 100081 p r china qianli xu department of mathematics yiyang teachers college yiyang 413049 p r china lihong huang department of applied mathematics hunan university changsha 410082 p r china weigao ge

on a conjecture of tian we study tian s alpha invariant in comparison with the alpha 1 invariant for pairs s d h consisting of a smooth surface s d of degree d in the projective three dimensional space and a hyperplane section h a conjecture of tian asserts that alpha s d h alpha 1 s d h we show that this is indeed true for d 4 the result is well known for d leqslant 3 and we show that alpha s d h alpha 1 s d h for d geqslant 8 provided that s d is general enough we also construct examples of s d for d 6 and d 7 for which tian s conjecture fails we provide a candidate counterexample for s 5

performance of two dual branch postdetection switch and stay combining schemes in correlated rayleigh and rician fading the performance of binary frequency shift keying bfsk and m ary frequency shift keying mfsk with dual branch postdetection switch and stay combining ssc in correlated rayleigh and rician fading is studied two postdetection ssc receivers are considered and the performances of noncoherent bfsk and mfsk are analyzed closed form expressions are derived for the average bit error rate ber of bfsk and mfsk with postdetection ssc in correlated rayleigh and rician fading optimum switching thresholds that minimize the average ber are obtained extensive monte carlo simulations are presented to test the validity of the analytical results the performance of dual branch postdetection ssc is compared with the performance of dual branch predetection ssc the effects of correlation fading parameter average fading power imbalance and switching threshold on the performances of postdetection ssc receivers are examined

on kahler einstein metrics on certain kahler manifolds with c1 m 0 on demontre qu il existe une metrique de kahler einstein sur une hypersurface de fermat a m dimensions de degre superieur a m 1

a further result on the oscillation of delay difference equations in this paper we are concerned with the delay difference equations of the form yn 1 yn pnyn k 0 n 0 1 2 where pn 0 and k is a positive integer we prove by using a new technique that n 0 i nn kpi ln i nn k pi 1 sign i nn k pi i n 1n k pi ln i n 1n k pi 1 sign i n 1n k pi guarantees that all solutions of equation oscillate which improves many previous well known results in particular our theorems also fit the case where n 1i n kpi kk 1 k 1 k 1 in addition we present a nonoscillation sufficient condition for equation

oscillation theorems for neutral difference equations abstract oscillation criteria for first order difference equations of the form x n q n x n 0 and x n px n k q n x n f n are established here xn xn 1 xn is the forward difference operator 1 p is a real number and k are nonnegative integers qn fn are sequences of nonnegative real numbers and r r is continuous oscillation criteria for nonlinear difference equations of sublinear as well as superlinear type are also established

a second order nonlinear difference equation oscillation and asymptotic behavior discrete analogues are investigated for well known results on oscillation growth and asymptotic behavior of solutions of y q t y 0 for q t 0 and for q t 0 the analogue of atkinson s oscillation criterion is shown to be true for 2yn 1 qn yn 0 but the analogue for atkinson s nonoscillation criterion is shown to be false

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

oscillation of nonlinear delay difference equations abstract in this paper two interesting oscillation criteria are obtained for all solutions of the nonlinear delay difference equations of the form yn 1 yn pnf yn k 0 n 0 1 2 some applications are given to demonstrate the advantage of results obtained in this paper our results also improve the existing results in the literature even when applying them to the linear delay difference equation

asymptotic behavior results for nonlinear neutral delay difference equations abstract this paper is concerned with the nonlinear neutral delay difference equation x n c n x n m p n f x n k 0 n n n 0 where is the forward difference operator defined by x n x n 1 x n c n is a sequence of real numbers p n is a positive sequence f c r r m and k are positive integers n 0 is a nonnegative integer and n n 0 n 0 n 0 1 n 0 2 sufficient conditions are obtained under which every solution of equation is bounded and tends to a constant as n our results improve and extend some known results one example is given to illustrate our results

explicit conditions for the oscillation of difference equations we obtain sufficient conditions for the oscillation of all solutions of the difference equation an 1 an i 1mpian ki 0 n 0 1 2 where the pi s are real numbers and the ki s are integers the conditions are given explicitly in terms of the pi s and the ki s

oscillation of nonlinear delay difference equations in this paper some new oscillation criteria are obtained for all solutions of the nonlinear delay difference equation of the form xn 1 xn pnf xn k 0 n 0 1 2 our results also improve some well known results in the literature even when applying them to the linear delay difference equation

asymptotic and oscillatory behavior of solutions of general nonlinear difference equations of second order abstract the asymptotic and oscillatory behavior of solutions of some general second order nonlinear difference equations of the form a n h y n 1 y n p n y n q n 1 f y n 1 0 n z is studied oscillation criteria for their solutions when pn is of constant sign are established results are also presented for the damped forced equation a n h y n 1 y n p n y n q n 1 f y n 1 e n n z examples are inserted in the text for illustrative purposes

oscillation theorems for second order advanced functional difference equations abstract we consider the second order advanced functional difference equation a n n p n g n 0 where a n 0 s n 0 1 a s p n 0 p n 0 g n n 1 g n is a monotone increasing integer sequence we obtain some new oscillation criteria through an appropriate riccati equation

thresholds valuations and k stability abstract let x be a normal complex projective variety with at worst klt singularities and l a big line bundle on x we use valuations to study the log canonical threshold of l as well as another invariant the stability threshold the latter generalizes a notion by fujita and odaka and can be used to characterize when a q fano variety is k semistable or uniformly k stable it can also be used to generalize volume bounds due to fujita and liu the two thresholds can be written as infima of certain functionals on the space of valuations on x when l is ample we prove that these infima are attained in the toric case toric valuations achieve these infima and we obtain simple expressions for the two thresholds in terms of the moment polytope of l

necessary and sufficient condition for oscillation of higher order nonlinear delay difference equations this paper is concerned with the oscillation of solutions of higher order nonlinear delay difference equations with forcing terms of the form nx t f t x t x t h t t i 0 1 where is the forward difference operator defined by x t x t 1 x t and mx t m 1x t m 1 a necessary and sufficient condition is established under which every solution x t is oscillatory when n is even and is either oscillatory or strongly monotone when n is odd a sufficient condition for the oscillation of solutions of neutral type delay difference equations is also obtained

integral averaging technique and oscillation of certain even order delay differential equations abstract using the generalized riccati technique and the averaging technique new oscillation criteria for certain even order delay differential equation 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

oscillation of delay difference equation we introduce some new techniques to analyze generalized characteristic equations to obtain several infinite sum sharp conditions for oscillation of the following delay difference equation xn 1 xn i 1mpi n xn ki 0 n 0 1 2 where pi n 0 and ki are positive integers

the oscillation of solutions of difference equations abstract using the solutions of a suitable inequality we shall discuss the asymptotic properties of the solutions of a generalized emden fowler difference equation in the absence of specified constant sign solutions of certain inequalities oscillatory behavior of all solutions of this equation are also deduced

comparison theorems for second order nonlinear difference equations abstract new oscillation results are obtained for the second order nonlinear difference equation r n f x n 1 g n x n 0 and its functional form r n f x n 1 g n x n 0 the role played by the argument n on the oscillation of the functional equation is explored in particular we characterize a class of sequences n which have a harmless effect on the oscillation of this type of equations some of our results generalize improve or unify known fundamental oscillation results for several particular cases of the above equations

riccati type transformations for second order linear difference equations abstract oscillation and comparison theorems for a linear homogeneous second order difference equation are proved by employing various equivalent non linear equations obtained by means of transformations analogous to the riccati transformation for ordinary differential equations

semicycles of solutions of nonlinear difference equations with several delays abstract in this paper we study the semicycles of oscillatory solutions of nonlinear difference equation with several delays where pi n is a real sequence with pi n 0 for all large n l mi kij are positive integers ij i 1 2 l j 1 2 mi are nonnegative real numbers with mij 1 ij 1 i 1 2 l upper bound of numbers of terms of semicycles are determined

oscillation theorems for certain neutral functional differential equations in this paper we shall offer sufficient conditions for the oscillation of all solutions to neutral functional differential equations of mixed type of the form dndtn x t ax t bx t q t x t g p t x t h 0 1 here p and q are periodic functions

complete characterizations of an oscillatory neutral difference equation this paper is concerned with a basic neutral type difference equation which depends on four parameters by means of the theory of envelopes we determine the exact range of the parameters for which the corresponding equation is oscillatory

asymptotic behavior of the solutions of second order difference equations abstract the asymptotic behavior of the solutions of the second order difference equation 2 x n n x n is considered

oscillation theorems for second order neutral difference equations abstract oscillation criteria for the second order difference equations of the form 2xn qnf xn r 0 and the neutral difference equation 2 xn pxn k qnf xn r 0 are established here xn xn 1 xn is the forward difference operator 1 p is a real number and k are nonnegative integers qn is a sequence of nonnegtive real numbers and f r r is continuous

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

when all solutions of x qi t x t ti t oscillate abstract in this paper the long term behavior of solutions to the equation in the title are examined where q i t and t i t are positive in particular it is shown that if lim inf t i 1 n t i t q i t 1 e all solutions oscillate about 0 infinitely often

focal boundary value problems for nonlinear difference equations ii on considere l unicite des solutions impliquant l existence des solutions de problemes aux valeurs limites de type focal pour l equation aux differences non lineaire d ordre n u t n f t u t u t n 1 ou f a r n r est continue et l equation u n 1 f t u 1 u n peut etre resolue pour u 1 comme fonction continue des variables t u 2 u u 1

multiple periodic solutions for superquadratic second order discrete hamiltonian systems abstract some multiplicity results are obtained for periodic solutions of the nonautonomous superquadratic second order discrete hamiltonian systems 2 u t 1 f t u t 0 t z by using critical point theory especially a three critical points theorem proposed by brezis and nirenberg

oscillation for partial difference equations with continuous variables in this paper we develop criteria for oscillation of the partial difference equation with continuous variables of the form au x a y b bu x a y cu x y b u x y d x y u x y e x y u x y f x y u x y 0 where d e f c r r r 0 a b and c are nonnegative constants a b and are positive constants several examples which dwell upon the importance of our results are also included

k stability and kahler einstein metrics we prove that if a fano manifold m is k stable then it admits a kahler einstein metric it affirms a longstanding conjecture for fano manifolds 2015 wiley periodicals inc

global asymptotic stability in a nonlinear nonautonomous difference equation with delays abstract in this paper some sufficient conditions are obtained for the global asymptotic stability of a nonlinear nonautonomous difference equation with delays and the works in 1 and the authors 2 are generalized

bounded and monotone properties of solutions of second order quasilinear forced difference equations abstract the authors consider the quasilinear difference equation p n 1 y n 1 q n y n r n and obtain results on the asymptotic behavior of solutions of including sufficient conditions for all solutions to be bounded or unbounded some results on the existence and behavior of nonincreasing solutions of are also obtained examples are inserted to illustrate the results

classification schemes for positive solutions of a second order nonlinear difference equation classification schemes for eventually positive solutions of a class of second order nonlinear difference equations are given in terms of their asymptotic magnitudes and necessary as well as sufficient conditions for the existence of these solutions are also provided

the semicycles of solutions of delay difference equations in this paper we study the semicycles of oscillatory solutions of the delay difference equation yn 1 yn pnyn k 0 where pn is is a sequence of nonnegative real numbers and k is a positive integer upper bound of numbers of terms of semicycles are determined