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

estimation of the mean of a multivariate normal distribution estimation of the means of independent normal random variables is considered using sum of squared errors as loss an unbiased estimate of risk is obtained for an arbitrary estimate and certain special classes of estimates are then discussed the results are applied to smoothing by use of moving averages and to trimmed analogs of the james stein estimate a suggestion is made for calculating approximate confidence sets for the mean vector centered at an arbitrary estimate

asymptotic properties of solutions of certain third order dynamic equations in this paper the well known oscillation criteria due to hille and nehari for second order linear differential equations will be generalized and extended to the third order nonlinear dynamic equation r 2 t r 1 t x d t d c d q t f x t 0 on time scale t where c 1 is a ratio of odd positive integers our results are essentially new even for third order differential and difference equations i e when t r and t n two examples of dynamic equations on different time scales are given to show the applications of our main results

small confidence sets for the mean of a spherically symmetric distribution summary suppose that x has a k variate spherically symmetric distribution with mean vector 0 and identity covariance matrix wepresent two spherical confidence sets for 0 both centred at a positive part stein estimator t x in the first we obtain the radius by approximating the upper a point of the sampling disiribution of iits x 9112 by the first two non zero terms of its taylor series about the origin we can analyse some of the properties of this confidence set and see that it performs well in terms of coverage probability volume and conditional behaviour in the second method we find the radius by using a parametric bootstrap procedure here even greater improvement in terms of volume over the usual confidence set is possible at the expense of having a less explicit radius function a real data example is provided and extensions to the unknown covariance matrix and elliptically symmetric cases are discussed

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

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

oscillation and asymptotic behavior of third order nonlinear retarded dynamic equations this paper is concerned with oscillation and asymptotic behavior of a class of third order nonlinear delay dynamic equations on an arbitrary time scale a new theorem is presented that improves a number of results reported in the literature examples are included to illustrate new results

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

projective modules over overrings of polynomial rings abstract let a be a commutative noetherian ring of dimension d and let p be a projective r a x 1 x l y 1 y m 1 f 1 f m module of rank r max 2 dim a 1 where f i a y i then i the natural map r gl r r el r 1 r k 1 r is surjective 3 8 ii assume f i is a monic polynomial then r 1 is an isomorphism 3 8 iii el 1 r p acts transitively on um r p in particular p is cancellative 3 12 iv if a is an affine algebra over a field then p has a unimodular element 3 13 in the case of laurent polynomial ring i e f i y i i ii are due to suslin 1977 12 iii is due to lindel 1995 4 and iv is due to bhatwadekar lindel and rao 1985 2

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

stably free modules over of rank dim are free we prove that for any finite dimensional ring r and n dim r 2 the group e n r x acts transitively on um n r x in particular we obtain that for any finite dimensional ring r all finitely generated stably free modules over r x of rank dim r are free this result was only known for noetherian rings the proof we give is short simple and constructive

oscillation of second order nonlinear neutral dynamic equations on time scales abstract this paper is concerned with the oscillation of second order nonlinear neutral dynamic equations of the form r t y t p t y t f t y t 0 on a time scale t by using a generalized riccati technique and integral averaging techniques we establish new oscillation results which handle some cases not covered by known criteria

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

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

simultaneous estimation of location parameters for sign invariant distributions estimation of p p 3 location parameters of a distribution of a p dimensional random vector x is considered under quadratic loss explicit estimators which are better than the best invariant one are given for a sign invariantly distributed random vector x the results depend only on the second and the third moments of x the generalizations to concave loss functions and to n observations are also considered additionally if the scale is unknown we investigate the estimators of the location parameters when the observation contains a residual vector

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

shrinkage estimators skorokhod s problem and stochastic integration by parts for a broad class of error distributions that includes the spherically symmetric ones we give a short proof that the usual estimator of the mean in a d dimensional shift model is inadmissible under quadratic loss when d 3 our proof involves representing the error distribution as that of a stopped brownian motion and using elementary stochastic analysis to obtain a generalization of an integration by parts lemma due to stein in the gaussian case

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

on the existence of unimodular elements and cancellation of projective modules over noetherian and non noetherian rings let r be a commutative ring of dimension d s r x or r x 1 x and p a finitely generated projective s module of rank r then p is cancellative if p has a unimodular element and r d 1 moreover if r dim s then p has a unimodular element and therefore p is cancellative as an application we prove that if r is a ring of dimension d of finite type over a prufer domain and p is a projective r x or r x 1 x module of rank at least d 1 then p has a unimodular element and is cancellative