the structure of general interference functions and applications this paper provides a theoretical framework for the analysis of interference coupled multiuser systems the fundamental behavior of such a system is modeled by interference functions defined by axioms ldquononnegativity rdquoscale invariance rdquo and ldquomonotonicity rdquo it is shown that every interference function has an interpretation as the optimum of a min max problem where the optimization is over a closed comprehensive positive coefficient set this provides new insight into the structure of general interference functions and its elementary building blocks conversely it is shown that every closed comprehensive positive set can be expressed as a level set of an interference function this shows a close connection between the analysis of interference functions and multiuser performance regions which are typically closed comprehensive the generality of this framework allows for a wide range of potential applications as an example we analyze the problem of interference balancing

strict convexity of the feasible log sir region the feasible log sir region is defined as a set of all signal to interference ratios sir expressed in logarithmic scale that can be supported in a wireless network by means of power control and with all users being active concurrently recently the feasible log sir region was shown to be a convex set which is a key ingredient in the development of some power control strategies for wireless systems in this paper under the assumption of a noiseless channel we strengthen these results by proving a necessary and sufficient condition for the feasible log sir region to be a strictly convex set the strict convexity property is of interest since it is closely related to the problem of the existence and uniqueness of a so called log sir fair power vector

a calculus for log convex interference functions the behavior of certain interference coupled multiuser systems can be modeled by means of logarithmically convex log convex interference functions in this paper we show fundamental properties of this framework a key observation is that any log convex interference function can be expressed as an optimum over elementary log convex interference functions the results also contribute to a better understanding of certain quality of service qos tradeoff regions which can be expressed as sublevel sets of log convex interference functions we analyze the structure of the qos region and provide conditions for the achievability of boundary points the proposed framework of log convex interference functions generalizes the classical linear interference model which is closely connected with the theory of irreducible nonnegative matrices perron frobenius theory we discuss some possible applications in robust communication cooperative game theory and max min fairness

a double sequence iteration process for fixed points of continuous pseudocontractions abstract let c be a bounded closed convex nonempty subset of a real hilbert space h the idea of a double sequence iteration is introduced and it is proved that a mann type double sequence iteration process converges strongly to a fixed point of a continuous pseudocontractive map t which maps c into c related results deal with the strong convergence of the iteration process to fixed points of nonexpansive maps

iterative process with errors of nonlinear equations involvingm accretive operators abstract let x be a uniformly smooth and uniformly convex banach space and t d t x x be an m accretive operator with the domain d t and the range r t for any given f x we prove that the mann and ishikawa type iterative sequences with errors converge strongly to the unique solution of the equation x tx f where t may not be lipschitz some related results deal with the convergence of mann and ishikawa type iterative sequences with errors for approximating a solution of a continuous dissipative operator equation

comparison of recent estimators of interclass correlation from familial data the asymptotic and finite sample properties of several recent estimators of interclass correlation are compared to more traditional estimators in the case of a variable number of siblings per family it is shown that karlin s family weighted pairwise estimator karlin cameron and williams 1981 proceedings of the national academy of science 78 2664 2668 is virtually equivalent to the ensemble estimator rosner donner and hennekens 1977 applied statistics 26 179 187 thus suggesting an estimator of the former s asymptotic variance further an estimator proposed by srivastava 1984 biometrika 71 177 185 is shown to be identical to the modified sib mean estimator konishi 1982 annals of the institute of statistical mathematics 34 505 515 when the sib sib correlation is estimated by the method of unweighted group means although the estimator due to srivastava has smaller asymptotic variance than the other two the gain in efficiency is slight for familial data both asymptotically and in finite samples

accounting for a decaying correlation structure in cluster randomized trials with continuous recruitment a requirement for calculating sample sizes for cluster randomized trials crts conducted over multiple periods of time is the specification of a form for the correlation between outcomes of subjects within the same cluster encoded via the within cluster correlation structure previously proposed within cluster correlation structures have made strong assumptions for example the usual assumption is that correlations between the outcomes of all pairs of subjects are identical uniform correlation more recently structures that allow for a decay in correlation between pairs of outcomes measured in different periods have been suggested however these structures are overly simple in settings with continuous recruitment and measurement we propose a more realistic continuous time correlation decay structure whereby correlations between subjects outcomes decay as the time between these subjects measurement times increases we investigate the use of this structure on trial planning in the context of a primary care diabetes trial where there is evidence of decaying correlation between pairs of patients outcomes over time in particular for a range of different trial designs we derive the variance of the treatment effect estimator under continuous time correlation decay and compare this to the variance obtained under uniform correlation for stepped wedge and cluster randomized crossover designs incorrectly assuming uniform correlation will underestimate the required sample size under most trial configurations likely to occur in practice planning of crts requires consideration of the most appropriate within cluster correlation structure to obtain a suitable sample size

the capacity region of the gaussian multiple input multiple output broadcast channel the gaussian multiple input multiple output mimo broadcast channel bc is considered the dirty paper coding dpc rate region is shown to coincide with the capacity region to that end a new notion of an enhanced broadcast channel is introduced and is used jointly with the entropy power inequality to show that a superposition of gaussian codes is optimal for the degraded vector broadcast channel and that dpc is optimal for the nondegraded case furthermore the capacity region is characterized under a wide range of input constraints accounting as special cases for the total power and the per antenna power constraints

relative efficiency of unequal cluster sizes in stepped wedge and other trial designs under longitudinal or cross sectional sampling background a cluster trial with unequal cluster sizes often has lower precision than one with equal clusters with a corresponding inflation of the design effect for parallel group trials adjustments to the design effect are available under sampling models with a single intracluster correlation design effects for equal clusters under more complex scenarios have appeared recently including stepped wedge trials under cross sectional or longitudinal sampling we investigate the impact of unequal cluster size in these more general settings results assuming a linear mixed model with an exchangeable correlation structure that incorporates cluster and subject autocorrelation we compute the relative efficiency re of a trial with clusters of unequal size under a size stratified randomization scheme as compared to an equal cluster trial with the same total number of observations if there are no within cluster time effects the re exceeds that for a parallel trial in general the re is a weighted average of the re for a parallel trial and the re for a crossover trial in the same clusters existing approximations for parallel designs are extended to the general setting increasing the cluster size by the factor 1 cv2 where cv is the coefficient of variation of cluster size leads to conservative sample sizes as in a popular method for parallel trials conclusion methods to assess experimental precision for single period parallel trials with unequal cluster sizes can be extended to stepped wedge and other complete layouts under longitudinal or cross sectional sampling in practice the loss of precision due to unequal cluster sizes is unlikely to exceed 12

combinatorial characterization of interference coupling in wireless systems we provide a combinatorial characterization of interference coupling in wireless systems with the intent of obtaining a better insight into interference coordination and management we introduce two bipartite graphs namely the power graph and interference graph we utilize these graphs and global dependency matrix containing only binary 0 and 1 entries to capture the effects of interference coupling in communication systems we show that the irreducibility of the global dependency matrix g is related to the connectivity of the power graph and the irreducibility of the matrix ggt is related to the connectivity of the interference graph we prove that for strictly positive and strictly log convex interference functions the irreducibility of the matrices g and ggt are necessary and sufficient conditions for the considered utility sets to be strictly convex in this case there exists a unique optimizer for the problem of maximizing the product of utilities we show that an interference balancing function is strictly log convex if and only if matrices g and ggt are irreducible we provide a simple yet comprehensive combinatorial characterization of interference coupled systems which abstracts away certain complexities of the physical layer

fair multiuser channel allocation for ofdma networks using nash bargaining solutions and coalitions in this paper a fair scheme to allocate subcarrier rate and power for multiuser orthogonal frequency division multiple access systems is proposed the problem is to maximize the overall system rate under each user s maximal power and minimal rate constraints while considering the fairness among users the approach considers a new fairness criterion which is a generalized proportional fairness based on nash bargaining solutions and coalitions first a two user algorithm is developed to bargain subcarrier usage between two users then a multiuser bargaining algorithm is developed based on optimal coalition pairs among users the simulation results show that the proposed algorithms not only provide fair resource allocation among users but also have a comparable overall system rate with the scheme maximizing the total rate without considering fairness they also have much higher rates than that of the scheme with max min fairness moreover the proposed iterative fast implementation has the complexity for each iteration of only o k sup 2 nlog sub 2 n k sup 4 where n is the number of subcarriers and k is the number of users

duality achievable rates and sum rate capacity of gaussian mimo broadcast channels we consider a multiuser multiple input multiple output mimo gaussian broadcast channel bc where the transmitter and receivers have multiple antennas since the mimo bc is in general a nondegraded bc its capacity region remains an unsolved problem we establish a duality between what is termed the dirty paper achievable region the caire shamai see proc ieee int symp information theory washington dc june 2001 p 322 achievable region for the mimo bc and the capacity region of the mimo multiple access channel mac which is easy to compute using this duality we greatly reduce the computational complexity required for obtaining the dirty paper achievable region for the mimo bc we also show that the dirty paper achievable region achieves the sum rate capacity of the mimo bc by establishing that the maximum sum rate of this region equals an upper bound on the sum rate of the mimo bc

the kullback leibler divergence and nonnegative matrices this correspondence establishes an interesting connection between the kullback leibler divergence and the perron root of nonnegative irreducible matrices in the second part of the correspondence we apply these results to the power control problem in wireless communications networks to show a fundamental tradeoff between fairness and efficiency a power vector is said to be efficient if it maximizes the overall network efficiency expressed in terms of an aggregate network utility function parameterized by some weight vector for two widely used examples of utility functions the correspondence identifies the unique weight vector for which a power vector is both efficient and max min fair in the sense that each communication link has the same quality of service these results also give rise to new saddle point characterizations of the perron root

multivariate extensions of mcnemar s test this article considers global tests of differences between paired vectors of binomial probabilities based on data from two dependent multivariate binary samples difference is defined as either an inhomogeneity in the marginal distributions or asymmetry in the joint distribution for detecting the first type of difference we propose a multivariate extension of mcnemar s test and show that it is a generalized score test under a generalized estimating equations gee approach univariate features such as the relationship between the wald and score tests and the dropout of pairs with the same response carry over to the multivariate case and the test does not depend on the working correlation assumption among the components of the multivariate response for sparse or imbalanced data such as occurs when the number of variables is large or the proportions are close to zero the test is best implemented using a bootstrap and if this is computationally too complex a permutation distribution we apply the test to safety data for a drug in which two doses are evaluated by comparing multiple responses by the same subjects to each one of them

a generalization of nash bargaining and proportional fairness to log convex utility sets with power constraints many solutions and concepts in resource allocation and game theory rely on the assumption of a convex utility set in this paper we show that the less restrictive assumption of a logarithmic hidden convexity is sometimes sufficient we consider the problems of nash bargaining and proportional fairness which are closely related we extend the nash bargaining framework to a broader family of log convex sets we then focus on the set of feasible signal to interference plus noise ratios sinrs for the cases of individual power constraints and a sum power constraint under the assumption of log convex interference functions we show how pareto optimality of boundary points depends on the interference coupling between the users finally we provide necessary and sufficient conditions for strict log convexity of the feasible sinr region

concave and convex interference functions general characterizations and applications many resource allocation problems can be studied within the framework of interference functions basic properties of interference functions are non negativity scale invariance and monotonicity in this paper we study interference functions with additional properties namely convexity concavity and log convexity such interference functions occur naturally in various contexts e g adaptive receive strategies robust power control or resource allocation over convex utility sets we show that every convex resp concave interference function can be expressed as a maximum resp minimum over a weighted sum of its arguments this analytical insight provides a link between the axiomatic interference framework and conventional interference models that are based on the definition of a coupling matrix we show how the results can be used to derive best possible convex concave approximations for general interference functions the results have further application in the context of feasible sets of multiuser systems convex approximations for general feasible sets are derived finally we show how convexity can be exploited to solve the problem of signal to interference plus noise ratio sinr constrained power minimization with super linear convergence

power and sample size requirements for gee analyses of cluster randomized crossover trials the cluster randomized crossover design has been proposed to improve efficiency over the traditional parallel cluster randomized design which often involves a limited number of clusters in recent years the cluster randomized crossover design has been increasingly used to evaluate the effectiveness of health care policy or programs and the interest often lies in quantifying the population averaged intervention effect in this paper we consider the two treatment two period crossover design and develop sample size procedures for continuous and binary outcomes corresponding to a population averaged model estimated by generalized estimating equations accounting for both within period and interperiod correlations in particular we show that the required sample size depends on the correlation parameters through an eigenvalue of the within cluster correlation matrix for continuous outcomes and through two distinct eigenvalues of the correlation matrix for binary outcomes we demonstrate that the empirical power corresponds well with the predicted power by the proposed formulae for as few as eight clusters when outcomes are analyzed using the matrix adjusted estimating equations for the correlation parameters concurrently with a suitable bias corrected sandwich variance estimator

choosing appropriate analysis methods for cluster randomised cross over trials with a binary outcome in cluster randomised cross over crxo trials clusters receive multiple treatments in a randomised sequence over time in such trials there is usual correlation between patients in the same cluster in addition within a cluster patients in the same period may be more similar to each other than to patients in other periods we demonstrate that it is necessary to account for these correlations in the analysis to obtain correct type i error rates we then use simulation to compare different methods of analysing a binary outcome from a two period crxo design our simulations demonstrated that hierarchical models without random effects for period within cluster which do not account for any extra within period correlation performed poorly with greatly inflated type i errors in many scenarios in scenarios where extra within period correlation was present a hierarchical model with random effects for cluster and period within cluster only had correct type i errors when there were large numbers of clusters with small numbers of clusters the error rate was inflated we also found that generalised estimating equations did not give correct error rates in any scenarios considered an unweighted cluster level summary regression performed best overall maintaining an error rate close to 5 for all scenarios although it lost power when extra within period correlation was present especially for small numbers of clusters results from our simulation study show that it is important to model both levels of clustering in crxo trials and that any extra within period correlation should be accounted for copyright 2016 john wiley sons ltd

iteration processes for nonlinear lipschitzian strongly accretive mappings in lp spaces abstract suppose x l p or l p 1 p 2 let t x x be a lipschitzian and strongly accretive map with constant k 0 1 and lipschitz constant l satisfying p p 1 l 2 p 2 p 1 1 k define s x x by sx tx x for arbitrary x 0 x the sequence x n n 1 is defined by x n 1 1 n x n n sy n y n 1 n x n n sx n n 0 where n n 1 n n 1 are two real sequences satisfying i 0 n 2 1 p 2 p 1 1 k k n l 2 n p 2 p l 2 1 w n w l 2 1 2 n p 2 p 1 1 k k n l 2 n p 1 2 l 2 1 w n w l 2 1 2 n 1 1 for each n ii 0 n min w w l 2 1 1 w 4 p 1 l 2 k p 1 2 l 2 w 1 for each n iii n n where w p 2 p 1 1 k p 2 p l 2 then x n 1 converges strongly to the unique solution of tx moreover if n 2 1 p 2 p 1 1 k k l 2 p 2 p l 2 1 w w l 2 1 2 p 2 p 1 1 k k l 2 p 1 2 l 2 1 w w l 2 1 2 1 1 and n for each n and some 0 min w w l 2 1 1 w 4 p 1 l 2 k p 1 2 l 2 w 1 then x n 1 q p n 2 x 1 q where q denotes the solution of tx and 1 4 1 p 2 p 1 1 k k l 2 p 2 p l 2 1 w w l 2 1 2 2 p 2 p 1 1 k k l 2 p 1 2 l 2 1 w w l 2 1 2 1 1 0 1 a related result details with the iterative approximation of lipschitz strongly pseudocontractive maps in x

the solution by iteration of nonlinear equations of hammerstein type let e be a real uniformly smooth banach space suppose n is a set valued locally accretive map with open domain d n c e k is a single valued map with domain d k c e such that im n c d k and k exists and is locally strongly accretive on im k if the equation f e x knx has a solution x e d n the strong convergence of iteration methods of the ishikawa and mann types to this solution is established related theorems deal with the iterative solution of nonlinear equations of the form f g x tx for the case where t is accretive explicit convergence rates are established miramare trieste november 1996 permanent address department of mathematics and computer science nnamdi azikiwe university p m b 5025 awka anambra state nigeria

iterative solution of nonlinear equations with strongly accretive operators abstract let e be a real banach space with a uniformly convex dual suppose t e e is a strongly accretive map with bounded range such that for each e the equation tx has a solution in e it is proved that each of the two well known fixed point iteration methods the mann and ishikawa iteration methods under suitable conditions converges strongly to a solution of the equation tx furthermore our method shows that such a solution is necessarily unique explicit error estimates are given our results resolve in the affirmative an open problem j math anal appl 151 no 2 1990 460 and generalize important known results

capacity regions and optimal power allocation for cdma cellular radio the capacity region of code division multiple access cdma is determined by the set of transmission rates combined with quality of service qos requirements which allow for a feasible power allocation scheme for n mobiles in a cellular network the geometrical and topological properties of the capacity region are investigated in the present paper for the case of unlimited and limited power respectively as a central result we show that the capacity region is convex by breaking the complicated topological structure into characteristic properties of its boundary and interior points each of interest in itself based on these results we furthermore investigate optimal power assignment schemes in the case that the demand of a community of users is infeasible weighted minimax and bayes solutions are explicitly determined as appropriate means to share the capacity of a cellular network in a reasonable and fair way

sample size and power for mcnemar s test with clustered data mcnemar s test is used to compare the distribution of two paired binary random variables when the data are clustered adjustment is needed to ensure that it is still a valid test this article presents two approximations for calculating the power and sample size for the adjusted mcnemar s test for clustered data working with a particular adjustment a simulation study is conducted to demonstrate the accuracy of these approximations the method is also applied to the design of a study involving positron emission tomography in detecting metastatic colorectal cancer and sensitivity of sample size computations to the design parameters are explored in this context

a sharpened condition for strict log convexity of the spectral radius via the bipartite graph abstract friedland 1981 10 showed that for a nonnegative square matrix a the spectral radius r e d a is a log convex functional over the real diagonal matrices d he showed that for fully indecomposable a log r e d a is strictly convex over d 1 d 2 if and only if d 1 d 2 c i for any c r here the condition of full indecomposability is shown to be replaceable by the weaker condition that a and a a be irreducible which is the sharpest possible replacement condition irreducibility of both a and a a is shown to be equivalent to irreducibility of a 2 and a a which is the condition for a number of strict inequalities on the spectral radius found in cohen et al 1982 8 such two fold irreducibility is equivalent to joint irreducibility of a a 2 a a and aa or in combinatorial terms equivalent to the directed graph of a being strongly connected and the simple bipartite graph of a being connected additional ancillary results are presented

derivatives of the perron root at an essentially nonnegative matrix and the group inverse of an m matrix abstract the results of the work in this paper can be divided into two major parts in the first part we develop formulae using group inverses of irreducible m matrices for the second order partial derivatives with respect to the entries of the perron root r at an essentially nonnegative and irreducible matrix a we use these formulae to obtain explicit expressions for the matrices 2 r a ij 2 and 2 r a ii jj and examine these expressions at some special cases when for example in addition to a being essentially nonnegative and irreducible a is row stochastic or a is of rank 1 and so on in the second major part of this paper we investigate for a singular and irreducible m matrix q a certain matrix g which is a positive diagonal scaling of the group inverse q of q we show for instance that z gz 0 for all real vectors z and use this result and some of the formulae obtained in the first part of the paper to provide a matrix theoretic proof of a theorem due to j e cohen math proc cambridge philos soc 86 1979 343 350 which states that for an essentially nonnegative and irreducible matrix a the perron root is a convex function of the main diagonal of a

sample size calculation for cluster randomized cross over trials the cluster randomized cross over design has been proposed in particular because it prevents an imbalance that may bring into question the internal validity of parallel group cluster trials we derived a sample size formula for continuous outcomes that takes into account both the intraclass correlation coefficient representing the clustering effect and the interperiod correlation induced by the cross over design

perron root minimization for interference coupled systems with adaptive receive strategies interference in multiuser systems is often characterized by a non negative and irreducible coupling matrix the maximum eigenvalue perron root of the weighted coupling matrix provides a single measure for the joint achievability of certain signal to interference ratios sir in this paper we address the more general case where the users are coupled by concave interference functions this corresponds to a system with adaptive receive strategies which minimize the interference received by each user a necessary and sufficient condition for feasibility is obtained by minimizing the perron root over the set of possible receive strategies this type of problem is directly related to the problem of weighted max min sir balancing this paper provides an analytical framework and an iterative algorithm that converges monotonically to a global optimum we also study an alternative approach based on a fixed point iteration this iteration is shown to converge to the global optimum if the sir targets lie on the boundary of the region

sample size determination for gee analyses of stepped wedge cluster randomized trials in stepped wedge cluster randomized trials intact clusters of individuals switch from control to intervention from a randomly assigned period onwards such trials are becoming increasingly popular in health services research when a closed cohort is recruited from each cluster for longitudinal follow up proper sample size calculation should account for three distinct types of intraclass correlations the within period the inter period and the within individual correlations setting the latter two correlation parameters to be equal accommodates cross sectional designs we propose sample size procedures for continuous and binary responses within the framework of generalized estimating equations that employ a block exchangeable within cluster correlation structure defined from the distinct correlation types for continuous responses we show that the intraclass correlations affect power only through two eigenvalues of the correlation matrix we demonstrate that analytical power agrees well with simulated power for as few as eight clusters when data are analyzed using bias corrected estimating equations for the correlation parameters concurrently with a bias corrected sandwich variance estimator

estimating efficacy in clinical trials with clustered binary responses when non compliance occurs in a clinical trial it may be of interest to supplement the intent to treat analysis with an analysis of the efficacy or biological effect of therapy sommer and zeger 1991 developed a method for estimating efficacy applicable to the case of a binary response variable and all or none compliance that assumes independent subject responses we extend this approach to accommodate within cluster correlations as may be expected in a cluster randomized design the method is illustrated using data from a controlled village randomized clinical trial conducted in indonesia to investigate the effect of vitamin a supplementation on mortality in children we find that within cluster correlations for these data are very small and that taking into account the clustering does not substantially affect inferences in this case additional calculations show that small within cluster correlations though larger than those found in the vitamin a data may have a large impact on efficacy inferences we also present the results of a simulation study that demonstrates the validness of the proposed approach for finite sample sizes

on the comparison of correlated proportions for clustered data mcnemar s test is often used to compare two proportions estimated from paired observations we propose a method extending this to the case where the observations are sampled in clusters the proposed method is simple to implement and makes no assumptions about the correlation structure we conducted a monte carlo simulation study to compare the size and power of the proposed method with a test developed earlier by eliasziw and donner in the presence of intracluster correlation the size of mcnemar s test can greatly exceed the nominal level the size of eliasziw and donner s test is also inflated for some correlation patterns the proposed method on the other hand is close to the nominal size for a variety of correlation patterns although it is slightly less powerful than eliasziw and donner s procedure the proposed method is a good alternative to eliasziw and donner s test when in practice little is known about the correlation pattern of the data

on the convexity of feasible qos regions the feasible quality of service qos region is the set of all qos vectors that can be provided to the users by means of power control with interference treated as noise in an interference limited scenario this set is determined by the perron root of some qos dependent nonnegative matrix in a previous work we showed that if the signal to interference ratio sir is a log convex function of the qos then the perron root is a log convex function this implies convexity of the feasible qos region in this correspondence we prove that the log convexity property is also necessary for the perron root to be convex for any choice of the path gain matrix interestingly a significantly less restrictive property is sufficient when the gain matrix is confined to be symmetric positive semidefinite

characteristic inequalities of uniformly convex and uniformly smooth banach spaces abstract let x be a real banach space with dual x and moduli of convexity and smoothness x e and x respectively for 1 p denotes the duality mapping from x into 2 x with gauge function t p 1 and j p denotes an arbitrary selection for j p let a r 0 0 t is strictly increasing and there exists c 0 such that t c x t 2 and f r 0 0 t is convex nondecreasing and there exists k 0 such that k x it is proved that x is uniformly convex if and only if there is a a such that x y p x p p j p x y x y x y x and x is uniformly smooth if and only if there is a f such that x y p x p p j p x y x y x y x where for given function f f x y is defined by f x y p 0 1 x ty x p t f t y x ty x dt these inequalities which have various applications can be regarded as general banach space versions of the well known polarization identity occurring in hilbert spaces

characterization of convex and concave resource allocation problems in interference coupled wireless systems this paper investigates the possibility of having convex or concave formulations of optimization problems for interference coupled wireless systems an axiomatic framework for interference functions proposed by yates in 1995 is used to model interference coupling in our paper the paper shows that under certain natural assumptions the exponential transformation is the unique transformation up to a positive constant for convexification of resource allocation problems for linear interference functions furthermore it is shown that under certain intuitive assumptions it is sufficient to check for the joint concavity convexity of sum of weighted functions of sinr inverse sinr with respect to s p es where p is the power vector of the users if we would like the resulting resource allocation problem to be concave convex this paper characterizes the largest class of utility functions and the largest class of interference functions respectively which allow a convex and concave formulation of a problem for interference coupled wireless systems it extends previous literature on log convex interference functions and provides boundaries on the class of problems in wireless systems which can be algorithmically tackled by convex optimization techniques

steepest descent approximations for accretive operator equations a necessary and sufficient condition is established for the strong convergence of the steepest descent approximation to a solution of equations involving quasi accretive operators defined on a uniformly smooth banach space miramare trieste march 1993

sample size calculation for stepped wedge and other longitudinal cluster randomised trials the sample size required for a cluster randomised trial is inflated compared with an individually randomised trial because outcomes of participants from the same cluster are correlated sample size calculations for longitudinal cluster randomised trials including stepped wedge trials need to take account of at least two levels of clustering the clusters themselves and times within clusters we derive formulae for sample size for repeated cross section and closed cohort cluster randomised trials with normally distributed outcome measures under a multilevel model allowing for variation between clusters and between times within clusters our formulae agree with those previously described for special cases such as crossover and analysis of covariance designs although simulation suggests that the formulae could underestimate required sample size when the number of clusters is small whether using a formula or simulation a sample size calculation requires estimates of nuisance parameters which in our model include the intracluster correlation cluster autocorrelation and individual autocorrelation a cluster autocorrelation less than 1 reflects a situation where individuals sampled from the same cluster at different times have less correlated outcomes than individuals sampled from the same cluster at the same time nuisance parameters could be estimated from time series obtained in similarly clustered settings with the same outcome measure using analysis of variance to estimate variance components copyright 2016 john wiley sons ltd

inequalities in banach spaces with applications well known that many problems in a hilbert space h are solved by applying the following identity ilax 1 a yi12 a412 1 llyl12 a 1 mx yl12 1 1 for all x y in h and 0 5 i 5 1 therefore one natural method to solve problems in a banach space xis to establish equalities and usually inequalities in xanalogous to 1 1 this is in fact shown by some authors recent work see for example reich l poffald and reich 2 lim 3 lim et af 4 prus and smarzewski 5 smarzewski 6 and reich 21 the purpose of this paper is to continue carrying out investigations in this line more precisely in section 3 we establish some inequalities analogous to 1 l in either uniformly convex or uniformly smooth banach spaces as corollaries we obtain new characteristics of p uniform convexity and q uniform smoothness of a banach space x via the functionals i lip and ii llq respectively we give in section 4 applications of the inequalities obtained in section 3 to fixed points of uniformly lipschitzian mappings and to best approximation theory