ser of threshold based hybrid selection maximal ratio combining in correlated nakagami fading the average symbol error rate of threshold based hybrid selection maximal ratio combining t hs mrc with correlated nakagami m fading channels is obtained for positive integer values of fading parameter m for a special correlation structure an accurate approximate analysis of t hs mrc in arbitrarily correlated nakagami m fading channels is presented by using a green matrix approximation the derivation allows different m ary linear modulation schemes we illustrate our methodology using m ary phase shift keying as an example

nonlinear multilevel models with an application to discrete response data summary a procedure is proposed for the analysis of multilevel nonlinear models using a linearization the case of log linear models for discrete response data is studied in detail nonlinear models arise in a number of circumstances notably when modelling discrete data in this paper we consider the multilevel nonlinear model as in linear multilevel models we shall consider the general case where any of the model coefficients can be random at any level and where the random parameters may also be specified functions of the fixed parameter estimates discussed by h goldstein r prosser and j rasbash in an as yet unpublished report in the next two sections we set out the model and define notation this is followed by a section on estimation and then some examples

analysis of hybrid selection maximal ratio combining in correlated nakagami fading we present an analysis of a hybrid selection maximal ratio combining diversity system over an evenly correlated slow frequency nonselective nakagami fading channel where the correlation coefficient between any pair of the diversity branch gain amplitudes is the same and all average branch signal to noise ratios snrs are equal in this system the l branches with the largest instantaneous snr out of n available branches are selected and combined using maximal ratio combining from the joint characteristic function cf of the instantaneous branch snrs we obtain an expression for the cf of the combiner output snr as a series of elementary cfs the expression can be conveniently used to obtain the symbol error probability of coherent detection of different m ary modulation schemes we illustrate our methodology using m ary phase shift keying as an example

strongly asymmetric discrete painleve equations the additive case we examine a class of discrete painleve equations which present a strong asymmetry these equations can be written as a system of two equations the right hand sides of which do not have the same functional form we limit here our investigation to two canonical families of the quispel roberts thompson qrt classification both of which lead to difference equations several new integrable discrete systems are identified

rational solutions of a differential difference kdv equation the toda equation and the discrete kdv equation in this paper a series of rational solutions are presented for a differential difference analogue of the kdv equation the toda equation and the discrete kdv equation these rational solutions are obtained using hirota s bilinear formalism and backlund transformations the crucial step is the use of nonlinear superposition formulae

casorati determinant solutions for the discrete painleve ii equation we present a class of solutions to the discrete painleve ii equation for particular values of its parameters it is shown that these solutions can be expressed in terms of casorati determinants whose entries are discrete airy functions the analogy between the tau function for the discrete pii and that of the discrete toda molecule equation is pointed out

discrete painleve equations and their appearance in quantum gravity we discuss an algorithmic approach for both deriving discrete analogues of painleve equations as well as using such equations to characterize similarity reductions of spatially discrete integrable evolution equations as a concrete example we show that a discrete analogue of painleve i can be used to characterize similarity solutions of the kac moerbeke equation it turns out that these similarity solutions also satisfy a special case of painleve iv equation in addition we discuss a methodology for obtaining the relevant continuous limits not only at the level of equations but also at the level of solutions as an example we use the wkb method in the presence of two turning points of the third order to parametrize at the continuous limit the solution of painleve i in terms of the solution of discrete painleve i finally we show that these results are useful for investigating the partition function of the matrix model in 2d quantum gravity associated with the measure exp t1z2 t2z4 t3z6

virtual branch analysis of symbol error probability for hybrid selection maximal ratio combining in rayleigh fading we derive analytical expressions for the symbol error probability sep for a hybrid selection maximal ratio combining h s mrc diversity system in multipath fading wireless environments with h s mrc l out of n diversity branches are selected and combined using maximal ratio combining mrc we consider coherent detection of m ary phase shift keying mpsk and quadrature amplitude modulation mqam using h s mrc for the case of independent rayleigh fading with equal signal to noise ratio averaged over the fading the proposed problem is made analytically tractable by transforming the ordered physical diversity branches which are correlated into independent and identically distributed i i d virtual branches which results in a simple derivation of the sep for arbitrary l and n we further obtain a canonical structure for the sep of h s mrc as a weighted sum of the elementary seps which are the seps using mrc with i i d diversity branches in rayleigh fading or equivalently the seps of the nondiversity single branch system in nakagami fading whose closed form expressions are well known we present numerical examples illustrating that h s mrc even with l spl lt n can achieve a performance close to that of n branch mrc

strongly asymmetric discrete painleve equations the multiplicative case we examine a class of multiplicative discrete painleve equations which may possess a strongly asymmetric form when the latter occurs the equation is written as a system of two equations the right hand sides of which have different functional forms the present investigation focuses upon two canonical families of the quispel roberts thompson classification which contain equations associated with the affine weyl groups d5 1 and e6 1 or groups appearing lower in the degeneration cascade of these two many new discrete painleve equations with strongly asymmetric forms are obtained

multilevel mixed linear model analysis using iterative generalized least squares summary models for the analysis of hierarchically structured data are discussed an iterative generalized least squares estimation procedure is given and shown to be equivalent to maximum likelihood in the normal case there is a discussion of applications to complex surveys longitudinal data and estimation in multivariate models with missing responses an example is given using educational data

surface critical exponents for models of polymer collapse and adsorption the universality of the theta and theta points the surface critical exponents are estimated both in ordinary and in special regimes from exact series expansions of up to 28 terms for a model of the theta point they are found to be in agreement with those derived for the theta point confirming the conjecture that the theta and theta points lie in the same universality class some essential features of the phase diagram for a self interacting self avoiding polymer in solution are also calculated to a higher degree of accuracy than previously obtained

meromorphic solutions of difference equations integrability and the discrete painleve equations the painlev e property is closely connected to differential equations that are integrable via related iso monodromy problems many apparently integrable discrete analogues of the painlev e equations have appeared in the literature the existence of sufficiently many finite order meromorphic solutions appears to be a good analogue of the painlev e property for discrete equations in which the independent variable is taken to be complex a general introduction to nevanlinna theory is presented together with an overview of recent applications to meromorphic solutions of difference equations and the difference and q difference operators new results are presented concerning equations of the form w z 1 w z 1 r z w where r is rational in w with meromorphic coefficients

a unified approach for calculating error rates of linearly modulated signals over generalized fading channels we present a unified analytical framework to determine the exact average symbol error rate ser of linearly modulated signals over generalized fading channels the results are applicable to systems employing coherent demodulation with maximal ratio combining multichannel reception the analyses assume independent fading paths which are not necessarily identically distributed in all cases the proposed approach leads to an expression of the average ser involving a single finite range integral which can be easily computed numerically in addition as special cases ser expressions for single channel reception are obtained these expressions reduce to well known solutions give alternative often simpler expressions for previous results or provide new formulas that are either closed form expressions or simple to compute numerically

on the algebraic structure of rational discrete dynamical systems we show how singularities shape the evolution of rational discrete dynamical sys tems the stabilisation of the form of the iterates suggests a description providing among other things generalised hirota form exact evaluation of the algebraic entropy as well as remarkable polynomial factorisation properties we illustrate the phenomenon explicitly with examples covering a wide range of models

rational solutions of the discrete time toda lattice and the alternate discrete painleve ii equation the yablonskii vorob ev polynomials yn t which are defined by a second order bilinear differential difference equation provide rational solutions of the toda lattice they are also polynomial tau functions for the rational solutions of the second painleve equation pii here we define two variable polynomials yn t h on a lattice with spacing h by considering rational solutions of the discrete time toda lattice as introduced by suris these polynomials are shown to have many properties that are analogous to those of the yablonskii vorob ev polynomials to which they reduce when h 0 they also provide rational solutions for a particular discretization of pii namely the so called alternate discrete pii and this connection leads to an expression in terms of the umemura polynomials for the third painleve equation piii it is shown that the backlund transformation for the alternate discrete painleve equation is a symplectic map and the shift in time is also symplectic finally we present a lax pair for the alternate discrete pii which recovers jimbo and miwa s lax pair for pii in the continuum limit h 0

miura transforms for discrete painleve equations a miura transformation is presented relating the discrete painleve ii equation to what plays the role of the modified d pii equation 34 in the painleve gambier classification the study of this transformation makes possible the derivation of the auto backlund transform of the discrete pii and allows one to obtain particular solutions to the latter moreover the use of this transformation makes it possible to construct a new mapping in the parameter space of pii the latter is a novel method for the construction of integrable discrete systems

bilinear discrete painleve equations based on the results of singularity confinement we derive bilinear expressions for the discrete painleve equations in these cases where a bilinear expression is not sufficient we obtain trilinear or higher multilinear expressions we show that the bilinear approach provides a natural framework for the derivation of backlund and schlesinger transforms for the discrete painleve equations

relative risk regression models with inverse polynomials the proportional hazards model assumes that the log hazard ratio is a linear function of parameters in the current paper we model the log relative risk as an inverse polynomial which is particularly suitable for modeling bounded and asymmetric functions the parameters estimated by maximizing the partial likelihood are consistent and asymptotically normal the advantages of the inverse polynomial model over the ordinary polynomial model and the fractional polynomial model for fitting various asymmetric log relative risk functions are shown by simulation the utility of the method is further supported by analyzing two real data sets addressing the specific question of the location of the minimum risk threshold

an extended quasi likelihood function summary wedderburn s original definition of quasi likelihood for generalized linear models is extended to allow the comparison of variance functions as well as those of linear predictors and link functions the relationship between generalized linear models and the use of transformations of the response variable is explored and the ideas are illustrated by three examples generalized linear models by allowing the full distributional assumption about the random component in the model to be replaced by a much weaker assumption in which only the first and second moments were defined in making this extension wedderburn widened the scope of generalized linear models in a way very similar to that of gauss when he replaced the assumption of normality in classical linear models by that of equal variance for generalized linear models with distributions in the exponential family likelihood ratio and score tests are used for testing hypotheses concerning nested subsets of covariates in the linear predictor and for assessing hypothesized link functions these methods are also applicable with wedderburn s form of quasi likelihood however neither of these methods is suitable for the comparison of different variance functions in this paper we introduce an extended quasi likelihood function which allows for the comparison of various forms of all the components of a generalized linear model i e the linear predictor the link function and the variance function we then apply the ideas to the analysis of several sets of data

performance analysis of generalized selection combining in generalized correlated nakagami m fading the exact performance of selection combining hybrid selection maximal ratio combining h s mrc and threshold based h s mrc in correlated nakagami m fading for positive integer values of fading parameter m is analyzed for a particular correlation structure this correlation model though not general is much more general than the equal correlation case and includes equal correlation as a special case the symbol error rate and outage probability of the three combining schemes are obtained by transforming the correlated branch gains into a set of conditionally independent branch gains numerical and simulation results show excellent agreement with theoretical results

infinite series representations of the bivariate rayleigh and nakagami m distributions despite the usefulness of the bivariate correlated nakagami m cumulative distribution function cdf in communications systems analysis an infinite series for the computation of this function is not known in this letter an infinite series representation of the bivariate cdf is derived bounds on the error resulting from truncation of the infinite series are also derived

approximate ser of h s mrc in nakagami fading with arbitrary branch power correlation an accurate approximation to the average symbol error rate of hybrid selection maximal ratio combining h s mrc in correlated nakagami m fading with positive integer values of m is obtained by using a green s matrix approximation this approach is valid for different m ary linear modulation schemes we illustrate our methodology using m ary phase shift keying as an example

modelling lactation using an inverse polynomial in a multilevel statistical model a method is outlined for modelling the relationship between breast milk production and time from delivery suitable for data collected longitudinally the relationship is described using inverse polynomials in a two level model the coefficients are specified as random variables across subjects allowing within and between subject variation to be separately estimated in a three level model day to day variation can also be separately estimated the model is illustrated using data from the chiang mai lactation project it allows the use of covariates that vary over time and provides a method of analysis that is compatible with the practical constraints on the timing of data collection that are inherent in longitudinal studies

a trivariate nakagami m distribution with arbitrary covariance matrix and applications to generalized selection diversity receivers this paper deals with a trivariate nakagami m distribution derived from the diagonal elements of a wishart matrix for this distribution infinite series representations for its probability density and cumulative distribution functions are derived having an arbitrary covariance matrix and integer order fading parameters moreover upper bounds on the error resulting from truncating the infinite series are obtained based on the derived formulas the performance of triple branch generalized selection combining gsc receivers is analyzed for this type of receivers the outage and the average bit error probability for a variety of modulation schemes are analytically obtained the performance of gsc receivers is compared to that of conventional selection and maximal ratio diversity schemes in order to check the accuracy and convergence of the derived formulas various performance evaluation results are presented and compared to equivalent simulation ones

ser and outage of threshold based hybrid selection maximal ratio combining over generalized fading channels the average symbol error rate and outage probability of threshold based hybrid selection maximal ratio combining t hs mrc in generalized fading environments are analyzed a t hs mrc combiner chooses the combined branches according to a predetermined normalized threshold and the strength of the instantaneous signal to noise ratio snr of each branch therefore the number of combined branches is a random variable rather than a fixed number as in conventional hybrid selection maximal ratio combining h s mrc using the moment generating function method a unified analysis of t hs mrc over various slow and frequency nonselective fading channels is presented both independent identically distributed and independent nonidentically distributed diversity branches are considered the derivation allows different m ary linear modulation schemes the theory is illustrated using coherent m ary phase shift keying in nakagami m fading as an example it is shown that previous published results are incorrect

on a unified approach to transformations and elementary solutions of painleve equations an algorithmic method is developed for investigating the transformation properties of second order equations of painleve type this method which utilizes the singularity structure of these equations yields explicit transformations which relate solutions of the painleve equations ii vi with different parameters these transformations easily generate rational and other elementary solutions of the equations the relationship between painleve equations and certain new equations quadratic in the second derivative of painleve type is also discussed

determinant structure of the rational solutions for the painleve ii equation two types of determinant representations of the rational solutions for the painleve ii equation are discussed by using the bilinear formalism one of them is a representation by the devisme polynomials and another one is hankel determinant representation they are derived from the determinant solutions of the kp hierarchy and toda lattice respectively

a fast scoring algorithm for maximum likelihood estimation in unbalanced mixed models with nested random effects on decrit un algorithme qui utilise des formules explicites pour l inverse et le determinant de la matrice de covariance donnee par la motte 1972 et evite l inversion des grandes matrices

bilinear discrete painleve ii and its particular solutions by analogy to the continuous painleve ii equation we present particular solutions of the discrete painleve ii d pii equation these solutions are of a rational and special function airy type our analysis is based on the bilinear formalism that allows us to obtain the tau function for d pii two different forms of bilinear d pii are obtained and we show that they can be related by a simple gauge transformation