residual maximum likelihood reml estimation of a neighbour model for field experiments a spatial analysis of field experiments is proposed which takes account of association between neighbouring plots the residual maximum likelihood reml method of patterson and thompson 1971 biometrika 58 545 554 is used to estimate parameters of a general neighbour model which can be expressed as an autoregressive moving average arma model three data sets are analysed to i highlight the need for a model selection procedure ii illustrate the differing results between incomplete block and neighbour analysis and the effect of including treated border plots in the design and iii illustrate the environmental variation within an experiment using prediction of trend

frequency measures on time scales with applications the concept of frequency measures of subsets of a time scale is introduced and the relevant properties are discussed then frequent oscillation is defined to strengthen the classical concept of oscillation applications are shown by deriving oscillation criteria for first order dynamic equations on time scales

new kamenev type oscillation criteria for second order nonlinear advanced dynamic equations this paper presents a new kamenev type oscillation criterion for a second order advanced dynamic equation on a time scale two examples are included to show that the result obtained complements and improves those reported in the literature

oscillation criteria for a class of second order emden fowler delay dynamic equations on time scales abstract by means of riccati transformation technique we establish some new oscillation criteria for the second order emden fowler delay dynamic equations x t p t x t 0 on a time scale t here is a quotient of odd positive integers with p t real valued positive rd continuous functions defined on t to the best of our knowledge nothing is known regarding the qualitative behavior of these equations on time scales our results in this paper not only extend the results given in r p agarwal m bohner s h saker oscillation of second order delay dynamic equations can appl math q 13 1 2005 1 18 but also unify the oscillation of the second order emden fowler delay differential equation and the second order emden fowler delay difference equation

on inequalities of lyapunov for linear hamiltonian systems on time scales in this paper we establish several new lyapunov type inequalities for linear hamiltonian systems on an arbitrary time scale t when the end points are not necessarily usual zeroes but rather generalized zeroes which generalize and improve all related existing ones including the continuous and discrete cases

existence of nonoscillatory solutions to neutral dynamic equations on time scales abstract in this paper we give an analogue of the arzela ascoli theorem on time scales then we establish the existence of nonoscillatory solutions to the neutral dynamic equation x t p t x g t f t x h t 0 on a time scale to dwell upon the importance of our results three interesting examples are also included

quasi complete latin squares construction and randomization summary a general method for constructing quasi complete latin squares based on groups is given this method leads to a relatively straightforward way of counting the number of inequivalent quasi complete latin squares of side at most 9 randomization of such designs is discussed and an explicit construction for valid randomization sets of quasicomplete latin squares whose side is an odd prime power is given it is shown that contrary to common belief randomization using a subset of all possible quasicomplete latin squares may be valid while that using the whole set is not

majorization and d optimality for complete block designs under correlations summary d optimal designs are derived in the complete block design set up for comparing treatments using ordinary least squares estimators when the experimental errors are correlated it is shown that the d optimal design is different from the weakly universally optimal design of morgan and chakravarti and kiefer and wynn

optimality criteria in survey sampling summary this paper attempts to fill the gap between some different and conflicting approaches in survey sampling based on a fixed population regression type model a class of optimality criteria similar to those well accepted in optimal experimental design theory is introduced the minimax and superpopulation approaches in survey sampling turn out to correspond to two extreme criteria in the proposed class this helps understand the role of randomization the strategy of simple random sampling with sample mean and the rao hartley cochran strategy are shown to be criterion robust

a catalogue of efficient neighbour designs with border plots we consider designs in linear blocks with border plots in which a treatment may affect the response on the two adjacent plots three series of designs are given i neighbour balanced designs in complete blocks ii neighbour balanced designs in blocks each of which lacks one treatment iii partially neighbour balanced designs in few complete blocks complete methods of construction and randomization are given optimality properties are discussed

oscillation of second order damped dynamic equations on time scales abstract the study of dynamic equations on time scales has been created in order to unify the study of differential and difference equations the general idea is to prove a result for a dynamic equation where the domain of the unknown function is a so called time scale which may be an arbitrary closed subset of the reals this way results not only related to the set of real numbers or set of integers but those pertaining to more general time scales are obtained in this paper by employing the riccati transformation technique we will establish some oscillation criteria for second order linear and nonlinear dynamic equations with damping terms on a time scale t our results in the special case when t r and t n extend and improve some well known oscillation results for second order linear and nonlinear differential and difference equations and are essentially new on the time scales t h n h 0 t q n for q 1 t n 2 etc some examples are considered to illustrate our main results

oscillation criteria for second order nonlinear neutral variable delay dynamic equations abstract this paper is concerned with oscillation of second order nonlinear neutral dynamic equations on time scales with a variable delay by using the generalized riccati technique and integral averaging technique new oscillation criteria are obtained for all solutions of the equation some results extend known results for difference equations when the time scale is the set z of positive integers and for differential equations when the time scale is r several examples are given to illustrate the results of the paper

oscillation criteria for second order nonlinear neutral delay dynamic equations in this paper we will establish some oscillation criteria for the second order nonlinear neutral delay dynamic equation r t y t p t y t f t y t 0 on a time scale t here 0 is a quotient of odd positive integers with r t and p t real valued positive functions defined on t to the best of our knowledge nothing is known regarding the qualitative behavior of these equations on time scales so this paper initiates the study

neighbour balanced block designs for correlated errors summary the paper considers a simple block model and assumes that the errors within the blocks are correlated following a stationary first order autoregressive process gill shukla 1985 consider this model but they restrict the set of the competing designs to a small subset of the possible designs the binary and equireplicate designs in the present paper we show that the designs determined by gill shukla are optimal over all possible designs for positive correlations and that neighbour balanced designs in general are highly efficient

positive solutions of second order half linear dynamic equations on time scales necessary and sufficient conditions are obtained for the existence of positive solutions of a second order half linear dynamic equation on time scales relations between this equation and an advanced type dynamic equation are also discussed

numerical oscillation and non oscillation for differential equation with piecewise continuous arguments of mixed type the paper deals with the oscillation and non oscillation of the runge kutta methods for a differential equation with piecewise continuous arguments of mixed type the conditions of the oscillation and non oscillation for the runge kutta method are obtained it is proved that oscillation of the analytic solution is not preserved by the runge kutta method under any conditions the conditions under which the non oscillation of analytic solutions is preserved by the runge kutta method are obtained some numerical experiments are given

asymptotic behavior of solutions of a third order nonlinear dynamic equation on time scales in this paper we will establish some sufficient conditions which guarantee that every solution of the third order nonlinear dynamic equation c t a t x t q t f x t 0 t t0 oscillates or converges to zero

oscillation criteria for second order matrix dynamic equations on a time scale we obtain oscillation criteria for a second order self adjoint matrix differential equation on a measure chain in terms of the eigenvalues of the coefficient matrices and the graininess function we illustrate our results with some nontrivial examples

some optimal designs for repeated measurements with autoregressive errors summary a number of individuals are observed at equally spaced time points different individuals behave independently but the observations on the same individual have the autocorrelation structure of a first order autoregressive process we are concerned with the optimal allocation of treatments assuming that at each time point a new selection must be made conditions for optimality are given for models without nuisance parameters and also with covariates and with block effects

contrasts under long range correlations the background of the paper is the empirical observation from a variety of subject areas that long range correlations appear to be much more frequent than has been previously assumed this includes high quality measurement series which are commonly treated as prototypes of i i d observations evidence is briefly cited in the paper it has already been shown elsewhere that long range dependence leads to results that can be qualitatively different from those obtained under short range dependence and in particular that long range dependence has drastic effects on the naive statistical treatment of absolute constants the natural question arising from this also of relevance for statistical practice is how the long range dependence affects the statistics for contrasts the main answer given in this paper is twofold

asymptotic behavior of solutions of third order nonlinear dynamic equations on time scales in this paper we will study asymptotic behavior of solutions to third order nonlinear dynamic equations on time scales of the form 1a 2 t 1a 1 t x d t a 1 d a 2 d q t f x t 0 by using the riccati technique and integral averaging technique two different types of criteria are established one of which extends some existing results and the other is new two examples of dynamic equations on different time scales are given to show the applications of the obtained results

one hundred years of the design of experiments on and off the pages of biometrika the earliest important papers on planned experiments in biometrika appeared within a year of each other in 1917 and 1918 roughly speaking one was concerned with design optimality and industrial experiments the other with agricultural trials and blocking we find this approximate division of the subject into two parts helpful in describing its development and growth as a result of the hostility between fisher and karl pearson much of the development of designs for agriculture in the 1920s and 1930s occurred off the pages of biometrika despite this we are able to trace a coherent history of the development of the subject from field trials and response surface methods to clinical trials and bayesian versions of design optimality

oscillation criteria of second order half linear dynamic equations on time scales in this paper by using the riccati transformation technique chain rule and inequalitya l lab l 1 l 1 b l 0 l 1 where a and b are positive constants we will establish some oscillation criteria for the second order half linear dynamic equation p t x d t c d q t x c t 0 t a b on time scales where c 1 is an odd positive integer our results not only unify the oscillation of half linear differential and half linear difference equations but can be applied on different types of time scales and improve some well known results in the difference equation case some examples are considered here to illustrate our main results

lyapunov inequality for linear hamiltonian systems on time scales the principal aim of this paper is to state and prove some lyapunov inequalities for linear hamiltonian system on an arbitrary time scale t so that the well known case of differential linear hamiltonian systems t r and the recently developed case of discrete hamiltonian systems t z are unified applying these inequalities a disconjugacy criterion for hamiltonian systems on time scales is obtained

existence of weak solutions of two point boundary value problems for second order dynamic equations on time scales abstract in this paper by using critical point theory we establish the existence of weak solutions of the two point boundary value problem for a second order dynamic equation on an arbitrary time scale t so that the well known case of differential dynamic systems t r and the recently developed case of discrete dynamic systems t z are unified to the best of our knowledge this is the first time that boundary value problems of dynamic equations on time scales have been dealt with by using critical point theory

robust designs for serially correlated observations summary the presence of serial correlation affects the design strategy of a statistical experiment it is the object of this paper to construct robust and efficient designs under such conditions special consideration is given to the two level factorial designs and weighing designs

universally optimal designs with blocksize p times 2 and correlated observations this paper addresses the problems of determination and construction of universally optimal designs in two dimensional blocks of size p x 2 assuming within block observations are correlated generalized least squares estimation of treatment contrasts is considered in four fixed block effects models i with fixed row and column effects ii with the row effects only iii with the column effects only and iv with neither row nor column effects for a general dependence structure and p 2 optimal designs for model i are found to coincide with the least squares optimal designs for general p models i iv and the within block correlation pattern described by the doubly geometric process interesting nonbinary block patterns are found for the universally optimal designs only for model iv for small positive correlations do binary blocks turn out to be best though binarity with respect to rows or columns is often required regardless of the model the conditions frequently coincide with those for optimal nested row column designs with uncorrelated errors one class of these designs is found to be optimal for at least some values of the correlation parameters under all four models and others are found to be optimal for particular models the exact form of the blocks for a universally optimum design is found to be quite sensitive to the blocksize and to the magnitude of the correlations under both models iii and iv

oscillation of nonlinear dynamic equations on time scales by means of riccati transformation techniques we establish some new oscillation criteria for the second order nonlinear dynamic equation p t x d t d q t f x s t 0 email protected a b on time scales in terms of the coefficients and the graininess function we illustrate our results with a nontrivial example

dynamic equations on time scales a survey the study of dynamic equations on time scales which goes back to its founder stefan hilger 1988 is an area of mathematics that has recently received a lot of attention it has been created in order to unify the study of differential and difference equations in this paper we give an introduction to the time scales calculus we also present various properties of the exponential function on an arbitrary time scale and use it to solve linear dynamic equations of first order several examples and applications among them an insect population model are considered we then use the exponential function to define hyperbolic and trigonometric functions and use those to solve linear dynamic equations of second order with constant coefficients finally we consider self adjoint equations and more generally so called symplectic systems and present several results on the positivity of quadratic functionals

a necessary and sufficient condition for oscillation of the sturm liouville dynamic equation on time scales in this paper we investigate oscillatory properties of the second order sturm liouville dynamic equation on a time scale r t x c t x 0 using the so called trigonometric transformation we establish a necessary and sufficient condition for oscillation of this condition is then used to derive an explicit oscillation criterion for this equation

the many proofs and applications of perron s theorem this paper chronicles the wide dispersal of perron s 1907 result on positive matrices into many fields of science the many proofs given during the last 93 years are categorized and critiqued including perron s original two proofs and a more natural proof is presented this simple to understand result of perron provides a unequaled vehicle for taking students on a tour of many applied areas with some depth

the leitmann schmitendorf advertising game with n players and time discounting abstract the extension of the leitmann schmitendorf advertising game to n players and positive time discounting is investigated we show that the strong time consistency of the open loop nash equilibrium is preserved as to optimal controls while the boundary solution is unaffected by the number of firms as well as discounting the inner solution depends on industry structure the fully symmetric version of the game allows us to identify the parameter regions wherein both solutions are sustainable

oscillations of delay differential equations with variable coefficients consider the delay differential equation t p t x t 0 where p t c t0 r and is a positive constant we show that every solution of this equation oscillates if tt p t dt 1 e for sufficiently large t and t0 p t exp tt p s ds 1 e 1 dt

oscillation criteria for second order linear differential equations abstract some oscillation criteria are given for the second order linear differential equation r t u t c t u t 0 where r t c 1 t 0 0 and c t c t 0 r these results improve oscillation criteria of wintner hartman kamenev yan and philos by using a generalized riccati transformation t a t r t u t u t f t where f c 1 is a given function on t 0 and a t exp 2 t f s ds

on lyapunov type inequalities for nonlinear dynamic systems on time scales in this paper by using elementary analysis we establish several new lyapunov type inequalities for the following nonlinear dynamic system on an arbitrary time scale t x d t email protected t x s t email protected t y t p 2y t y d t email protected t x s t q 2x s t email protected t y t when the end points are not necessarily usual zeros but rather generalized zeros which generalize and improve all related existing ones including the continuous and discrete cases

optimal repeated measurements designs for correlated observations and analysis by weighted least squares summary we consider a model for repeated measurements designs where the measurements on the same unit are assumed to be correlated with known correlation coefficient it is shown that latin squares with an additional balancing property are e optimal for the weighted least squares estimate

efficient block designs for settings with spatially correlated errors summary two dimensional block designs with block size p x 2 for v treatments are studied for situations in which the errors are spatially correlated conditions for universal optimality are given for two different error covariance structures the doubly geometric for which the correlations decay rapidly and the autonormal which exhibits a slower rate of decay in the special cases of p v for even v to obtain complete blocks and p 1 v 1 for odd v for nearly complete blocks numerical calculations establish that much smaller designs using only a fraction of the blocks required by universal optimality are also reasonably efficient a table of designs with v 30 is included

on a disconjugacy criterion for second order dynamic equations on time scales in this paper a sufficient condition for disconjugacy of second order differential equations on time scales is given for this purpose a mean value result on time scales is employed

some oscillation results for second order nonlinear dynamic equations of neutral type abstract in this paper some oscillation results for the second order nonlinear neutral delay dynamic equation r t y t p t y t q t y t sgn y t 0 are obtained on a time scale t under the assumption t 0 1 r t 1 t where 0 is a quotient of odd positive integers

the nash solution of an advertising differential game generalization of a model by leitmann and schmitendorf in this paper the differential game between two profit maximizing firms considered in 15 is extended to the case of general concave effectiveness functions of advertising for this model a phase portrait analysis of nash equilibrium solutions is carried out providing qualitative insights into the structure of optimal advertising rates applying this analysis to the leitmann schmitendorf advertising model we obtain more detailed information on the structure of the optimal controls moreover nash optimal solutions are derived for infinite duration of the game

oscillation of second order nonlinear neutral delay dynamic equations on time scales in this paper some sufficient conditions for oscillation of the second order nonlinear neutral delay dynamic equation r t y t p t y t f t y t 0 on a time scale t are established here 1 is an odd positive integer with r t and p t are rd continuous functions defined on t our results as a special case when t r and t n involve and improve some well known oscillation results for second order neutral delay differential and difference equations when t hn and t qn t t qk k n q 1 i e for generalized neutral delay difference and q neutral delay difference equations our results are essentially new and also can be applied on different types of time scales e g t n2 t2 t n and t tn tn n n0 where tn is the set of harmonic numbers some examples illustrating our main results are given

oscillation criteria for second order nonlinear delay dynamic equations abstract in this paper we consider the second order nonlinear delay dynamic equation r t x t p t f x t 0 on a time scale t by employing the generalized riccati technique we will establish some new sufficient conditions which ensure that every solution oscillates or converges to zero the obtained results improve the well known oscillation results for dynamic equations and include as special cases the oscillation results for differential equations some applications and examples are considered to illustrate the main results

some new dynamic inequalities on time scales in this paper based on some known dynamic inequalities we investigate certain new dynamic inequalities on time scales which provide explicit bounds on unknown functions our results unify and extend some continuous inequalities and their corresponding discrete analogues