moment based criteria for determining bioequivalence summary in comparative bioavailability trials the focus in the determination of bioequivalence has been to show that the average bioavailability on a test formulation is similar to the average bioavailability on a reference formulation it has been argued that this is not adequate to insure interchangeability of formulations we introduce a moment based criterion which ensures interchangeability for at least a certain proportion of the population an algorithm is given for determining the constant in the criterion an example is used to illustrate how the criterion may be applied

interval criteria for forced oscillation with nonlinearities given by riemann stieltjes integrals abstract we study the oscillation of second order forced differential equations with nonlinearity given by a riemann stieltjes integral of the form p t x q t x 0 b r t s x t s sgn x t d s e t where b 0 c 0 b is strictly increasing such that 0 0 1 b p q e c 0 with p t 0 r c 0 0 b and 0 b r is nondecreasing interval oscillation criteria of the el sayed type and the kong type are obtained as a special case the work in this paper unifies and improves the existing results in the literature for equations with a finite number of nonlinear terms we also extend our results to equations with delays

oscillation criteria for second order forced ordinary differential equations with mixed nonlinearities abstract we present new oscillation criteria for the second order forced ordinary differential equation with mixed nonlinearities p t x q t x i 1 n q i t x i sgn x e t where p t q t q i t e t c 0 p t is positive and differentiable 1 m 1 m 1 n no restriction is imposed on the forcing term e t to be the second derivative of an oscillatory function when n 1 our results reduce to those of el sayed m a el sayed an oscillation criterion for a forced second order linear differential equation proc amer math soc 118 1993 813 817 wong j s w wong oscillation criteria for a forced second linear differential equations j math anal appl 231 1999 235 240 sun ou and wong y g sun c h ou j s w wong interval oscillation theorems for a linear second order differential equation comput math appl 48 2004 1693 1699 for the linear equation nazr a h nazr sufficient conditions for the oscillation of forced super linear second order differential equations with oscillatory potential proc amer math soc 126 1998 123 125 for the superlinear equation and sun and wong y g sun j s w wong note on forced oscillation of nth order sublinear differential equations j math anal appl 298 2004 114 119 for the sublinear equation

forced oscillation of nth order nonlinear differential equations the oscillation criteria result of agarwal and grace appl math lett 13 2000 53 is generalized to the following nonlinear equations of the formx n t email protected i 1 n 1a ix i t q t f x t e t email protected n where email protected c 1 r r and f is a strictly monotone increasing function email protected c r r a 1 a n 1 are real constants

a dissipative singular sturm liouville problem with a spectral parameter in the boundary condition abstract in the hilbert space l w 2 a b we consider nonselfadjoint singular sturm liouville boundary value problem with two singular end points a and b in limit circle cases at a and b and with a spectral parameter in the boundary condition the approach is based on the use of the maximal dissipative operator and the spectral analysis of this operator is adequate for boundary value problem we construct a selfadjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations which make it possible to determine the scattering matrix of the dilation we also construct a functional model of the maximal dissipative operator and define its characteristic function in terms of solutions of the corresponding sturm liouville equation on the basis of the results obtained regarding the theory of the characteristic function we prove theorems on completeness of the system of eigenvectors and associated vectors of the maximal dissipative operator and sturm liouville boundary value problem

oscillation of solutions of second order mixed nonlinear differential equations under impulsive perturbations new oscillation criteria are obtained for second order forced mixed nonlinear impulsive differential equations of the form r t f a x q t f a x k 1nq k t f b k x e t t q ix q i a ix q i x q i b ix q i where f c s s c 1s and b 1 b 2 b m a b m 1 b n 0 if a 1 and the impulses are dropped then the results obtained by sun and wong y g sun j s w wong oscillation criteria for second order forced ordinary differential equations with mixed nonlinearities j math anal appl 334 2007 549 560 are recovered examples are given to illustrate the results

a note on nasr s and wong s papers abstract in the case of oscillatory potentials we give sufficient conditions for the oscillation of the forced nonlinear second order differential equations with delayed argument in the form x t q t x t sgn x t f t in the linear 1 and the superlinear 1 cases

on the oscillation of hill s equations under periodic forcing abstract we use the floquet theory of the hill s equation to prove the conjecture that all solutions of the second order forced linear differential equation y c sin t y cos t are oscillatory on 0 for all c 0

interval oscillation criteria for a forced second order nonlinear ordinary differential equations with oscillatory potential new oscillation criteria established in the paper for a forced second order nonlinear equation with oscillatory potential p t y t q t f y t g t are different from the most known ones in the sense that they are based on the information only on a sequence of subintervals of t 0 rather than on the whole half line our results are of a high degree of generality and sharper than some previous results and handle the cases which are not covered by known results moreover these results can also be applied to the extreme such as t 0 q s ds

oscillation criteria for second order nonlinear differential equations involving general means consider the second order nonlinear differential equation e y a t f y 0 where a t c 0 f y c1 f y 0 and yf y 0 for y 0 furthermore f y also satisfies either a superlinear or a sublinear condition which covers the nonlinear function f y y y 1 with 1 and 0 1 respectively commonly known as the emden fowler case here the coefficient function a t is allowed to be negative for arbitrarily large values of t kamenev type oscillation criteria involving integral averages for the linear equations l y a t y 0 are extended to the nonlinear equation e by using more general means the results extend similar results on general means by philos for the linear equation l and also results based upon kamenev s integral averaging method concerning the nonlinear equation e

note on forced oscillation of nth order sublinear differential equations in the case of oscillatory potentials we establish an oscillation theorem for the forced sublinear differential equation x n q t x sgnx e t t t0 no restriction is imposed on the forcing term e t to be the nth derivative of an oscillatory function in particular we show that all solutions of the equation x t sint x sgnx mt cost t 0 0 2 1 this provides an analogue of a result of nasr proc amer math soc 126 1998 123 for the forced superlinear equation and answers a question raised in an earlier paper j s w wong siam j math anal 19 1988 673

on the perturbation determinants of a singular dissipative boundary value problem with finite transmission conditions abstract in this paper a singular dissipative boundary value problem with finite transmission conditions is investigated using livsic s theorem it is proved that the system of all eigen and associated functions of this problem is complete in the hilbert space

oscillation criteria for nth order forced functional differential equations in this paper by using the method of general means some new oscillation criteria for forced functional differential equations of the form x n t i 1n 1aix i t q t fxg t e t t0 0 are established where ai are real constants q t f t e t and g t are real continuous functions xf x 0 whenever x 0 and limt g t

oscillation criteria for second order neutral equations with distributed deviating argument abstract we present new oscillation criteria for the second order neutral delay differential equations with distributed deviating argument r t x t z t a b p t f x g t d 0 t t 0 where z t x t q t x t new oscillation criteria are established which are based on a class of new functions t s l defined in the sequel our results are sharper than some previous results which can be seen by the example at the end of this paper

forced oscillation of nth order functional differential equations abstract in this paper we investigate the oscillation of forced functional differential equations x n t a t f x q t e t when the forcing term e t is not required to be the nth derivative of an oscillatory function some new oscillatory criteria are given we also apply our technique to the forced neutral differential equation of the form x t cx t n a t x t b t x t e t c t f 1 x t d t f 2 x t where xf1 x 0 and xf2 x 0 for x 0 n 1 are nonnegative constants a t 0 b t 0 c t 0 d t 0 which includes the special case f1 x x sgn x f2 x x sgn x 1 and 1

the determinants of fourth order dissipative operators with transmission conditions abstract in this paper discontinuous non self adjoint differential operators in weyl s limit circle are studied we give the determinant of perturbation connected with the dissipative operator l generated by fourth order differential expression in l 2 i where discontinuity of operator is dealt with transmission conditions we obtain the green s function then using characteristic determinant we prove the completeness of the system of eigenfunctions and associated functions of this dissipative operator

interval oscillation theorems for asecond order linear differential equation interval oscillation criteria are given for the forced second order linear differential equation ly t p t y q t y t email protected 0 where p q are locally integrable functions and p t 0 for t 0 no restriction is imposed on t to be the second derivative of an oscillatory function as assumed by kartsatos 1 our results also allow both q and f to change sign in the neighborhood at infinity in particular we show that all solutions of y c sin t y t b cos t with b 0 are oscillatory for c 1 3448 this improves an estimate given by nasr 2 for the linear equation

the subject by formulation interaction as a criterion of interchangeability of drugs summary the primary purpose of a bioequivalence study is often to meet the regulatory requirement of similar average bioavailabilities this is usually done by estimating the mean difference between formulations in a crossover design however a small difference as well as a rather narrow confidence interval might be obtained also if the subjects respond differently to the formulations i e if there is a nonnegligible subject by formulation interaction hence it can be argued that this interaction should be part of the criterion of interchangeability to be able to study this interaction the ordinary simple crossover design is not appropriate instead a design where each formulation is administered at least twice to each subject is desirable such a design is obtained by repeating a simple crossover layout to evaluate the impact of the subject by formulation interaction the ratio of the variance components of the interaction and the error is estimated this approach is illustrated by an example two different formulations of a drug are considered bioequivalent if their average bioavailabilities are similar the assessment of this similarity is usually based on an interval estimate of the mean difference between the formulations however this focus on means is an insufficient criterion since a small difference as well as a rather narrow confidence interval might be obtained also if the formulation differences show a considerable variation among subjects the impact on drug interchangeability of this so called subject byformulation interaction is discussed for example by hwang et al 1978 in order to estimate this interaction they suggest a design where each formulation is administered at least twice to each subject in this paper we briefly discuss how to design and analyze a bioequivalence study in order to evaluate the impact of the subject by formulation interaction as a measure of the magnitude of the interaction we propose the ratio of the variance components of the interaction and the error the ideas of the paper are illustrated with data from a comparative study of two generic formulations of furosemide

asymptotics and oscillation of nth order nonlinear dynamic equations on time scales this paper is concerned with nth order nonlinear dynamic equations on time scales of the form r t x n 1 t i 0 k q i t i x i t 0 with n 2 by discussing the signs of ith order derivatives of eventually positive solutions for i 1 n 1 and using the generalized riccati technique and integral averaging technique we derive new criteria for oscillation and asymptotic behavior of the equation our results extend many existing results in the literature

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

interval criteria for oscillation of second order differential equations with mixed nonlinearities abstract in this paper we establish interval criteria for oscillation of the second order forced ordinary differential equations with mixed nonlinearities p t x q t x i 1 n q i t x i sgn x e t where p t q t qi t e t are continuous functions defined on 0 p t is positive and differentiable 1 m 1 m 1 n 0 n m 1 no restriction is imposed on the potentials q t qi t and e t to be nonnegative

sturmian theory for second order differential equations with mixed nonlinearities in the paper sturmian comparison theory is developed for the pair of second order differential equations first of which is the nonlinear differential equations 1 m t y s t y i 1 n q i t y i 1 y 0 with mixed nonlinearities 1 m 1 m 1 n and the second is the nonselfadjoint differential equations 2 k t x r t x p t x 0 under the assumption that the solution of eq 2 has two consecutive zeros we obtain sturm picone type and leighton type comparison theorems for eq 1 by employing the new nonlinear version of picone s formula that we derive wirtinger type inequalities and several oscillation criteria are also attained for eq 1 examples are given to illustrate the relevance of the results

numerical methods for the evaluation of individual bioequivalence criteria the evaluation of individual bioequivalence ibe by bootstrap resampling using common statistical software for example sas is extremely time consuming in this article an estimation procedure that can be implemented in a high level language with the same degree of accuracy as sas is described the necessary parameter estimating equations under both least square lse and restricted maximum likelihood reml methods are given the algorithms used to numerically compute these values are outlined and tested in fortran on several simulated data sets and shown to reproduce sas results with at least 10 3 precision more importantly the reml bootstrap algorithm reduces the time taken in sas by a factor of 20 secondary results indicate that lse and reml parameter estimates are similar for mild unbalancedness proc mixed with unstructured un and compound symmetry heterogeneous csh variance structures give the same results except when the subject by treatment interaction variance sigma 2 d is 0 in which case csh significantly overestimates sigma 2 d and underestimates the within treatment variances it is concluded that bootstrap evaluation of ibe is efficiently done using either the lse or reml algorithm in fortran

oscillation criteria for a forced mixed type emden fowler equation with impulses some oscillation criteria for a forced mixed type emden fowler equation with impulses are given when the impulses are dropped our results extend those of sun and meng y g sun f w meng interval criteria for oscillation of second order differential equations with mixed nonlinearities appl math comput 15 2008 375 381 sun and wong y g sun j s w wong oscillation criteria for second order forced ordinary differential equations with mixed nonlinearities j math anal appl 334 2007 549 560 for second order forced ordinary differential equation with mixed nonlinearities nasr a h nasr sufficient conditions for the oscillation of forced superlinear second order differential equations with oscillatory potential proc am math soc 126 1998 123 125 yang q yang interval oscillation criteria for a forced second order nonlinear ordinary differential equations with oscillatory potential appl math comput 135 2003 49 64 for forced superlinear emden fowler equation kong q kong interval criteria for oscillation of second order linear differential equations j math anal appl 229 1999 483 492 for unforced second order linear differential equations and wong j s w wong oscillation criteria for a forced second order linear differential equation j math anal appl 231 1999 235 240 for forced second order linear differential equation

oscillation of higher order forced nonlinear differential equations abstract some new criteria are established for the oscillation of higher order forced nonlinear differential equations by introducing a class of new auxiliary functions no restriction is imposed on the forcing term as is generally assumed with the help of the new auxiliary functions the main results in this paper are different from those in the paper y g sun s h saker forced oscillation of higher order nonlinear differential equations appl math comput 173 2006 1219 1226 and are more effective than many existing results

on oscillatory solutions of certain forced emden fowler like equations abstract we give a constructive proof of existence to oscillatory solutions for the differential equations x t a t x t sign x t e t where t t 0 1 and 1 that decay to 0 when t as o t for 0 as close as desired to the critical quantity 2 1 for this class of equations we have lim t e t 0 where e t 0 and e t e t throughout t 0 we also establish that for any and any negative valued e t o t as t the differential equation has a negative valued solution decaying to 0 at as o t in this way we are not in the reach of any of the developments from the recent paper c h ou j s w wong forced oscillation of nth order functional differential equations j math anal appl 262 2001 722 732

definition of individual bioequivalence occasion to occasion versus mean switchability two moment based scaled definitions of individual bioequivalence are discussed based on a mixed effects linear model their evaluations respectively lead to an unweighted theta 11 and a parametric theta 15 metric the two metrics are estimated with respect to study design and two estimation methods results show that the two ibe metrics perform equivalently in the fully replicated design in the semi replicated design the definition of theta 11 may not be valid while the evaluation of theta 15 results in a reduction of the weights in the mean difference and switchability components of the metric percentage rejection rates in the latter design indicate that theta 11 is more conservative than theta 15 this is because there is an increase of about 15 per cent in the producer risk in theta 11 relative to theta 15 compared to a 7 per cent increase in the consumer risk in theta 15 relative to theta 11 a further disadvantage of the design is that there is a 33 per cent loss in the subject by treatment variance efficiency which is reflected in a similar amount of decreased sensitivity to departures from perfect bioequivalence even when more subjects are used to equalize the number of exposure occasions in the two designs it is concluded that a mean switchability criterion may be more appropriate from an interpretability perspective the bootstrap resampling method used to evaluate individual bioequivalence based on theta 11 may need to be bias corrected and that the semi replicated design should be used cautiously

on forced oscillation of higher order functional differential equations abstract in this paper we establish some new oscillation criteria for the following forced functional differential equation x n t q t x t 1 x t e t t t 0 no restriction is imposed on the forcing term e t to be the n th derivative of an oscillatory function two cases have been taken into consideration i q t 0 1 and t t t ii q t changes its sign 0 t t t the main results generalize some existing results and can be applied to the oscillation problem that are not covered in the literature

forced oscillation of higher order nonlinear differential equations some new criteria have been established for the oscillation of higher order forced nonlinear differential equation of the form l nx t q t f x t e t according to the smart choice of a class of new auxiliary functions no restriction is imposed on the forcing term e t as those assumed in a g kartsatos maintenance of oscillations under the effect of a periodic forcing term proc amer math soc 33 1972 377 383 a g kartsatos recent results on oscillation of solutions of forced and perturbed nonlinear differential equations of even order in stability of dynamical systems theory and applications marcel dekker new york 1977 pp 17 72 the main results of this paper are different from most known ones in the sense that they are given in the form lim inf t 0

oscillation criteria for second order forced elliptic differential equations with mixed nonlinearities in the case of oscillatory potentials we establish some oscillation criteria for the second order forced elliptic differential equation with mixed nonlinearities div a x y c x y e x when n 1 our results extend those of kong q kong interval criteria for oscillation of second order linear ordinary differential equations j math anal appl 229 1999 258 270 for certain second order linear ordinary differential equations sun and meng y g sun f w meng interval criteria for oscillation of second order differential equations with mixed nonlinearities appl math comput 2007 doi 10 10 1016 j amc 2007 08 042 for certain second order differential equations with mixed nonlinearities yang q yang interval oscillation criteria for a second order nonlinear ordinary differential equations with oscillatory potential appl math comput 135 2003 49 64 for a forced second order superlinear ordinary differential equation when n 2 our theorems are more general and improve the oscillation criteria of zhuang r k zhuang annual oscillation criteria for second order nonlinear elliptic differential equations j comput anal math 2007 doi 10 1016 j cam 2007 06 031 for a forced second order undamped elliptic differential equation

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

integral averages and oscillation of second order nonlinear differential equations new oscillation criteria are given for the second order sublinear differential equation left a left t right psi left x left t right right x left t right right q left t right f left x left t right right 0 t geqslant t 0 0 where a c 1 t 0 is a nonnegative function f c with x 0 xf x x 0 for x 0 f have continuous derivative on 0 with f x x03c8 x 0 for x 0 and q c t 0 has no restriction on its sign this oscillation criteria involve integral averages of the coefficients q and a and extend known oscillation criteria for the equation x t q t x t 0

oscillation properties for even order neutral equations with distributed deviating arguments in this paper we establish several oscillatory criteria for even order neutral differential equations with distributed deviating arguments which generalize and improve some known results