on the capacity formula for multiple input multiple output wireless channels a geometric interpretation the capacity of multiple input multiple output mimo wireless channels is computed for ricean channels the novelty is a geometrical ray tracing interpretation of the mimo channel capacity formula to find array geometries which greatly enhance channel capacity compared to single input single output siso systems

uniqueness theorems for sub harmonic functions with applications to operator theory we obtain uniqueness theorems for harmonic and subharmonic functions of a new type they lead to new analytic extension criteria and new conditions for stability of operator semigroups in banach spaces with fourier type

normal families new perspectives this paper surveys some surprising applications of a lemma characterizing normal families of meromorphic functions on plane domains these include short and efficient proofs of generalizations of i the picard theorems ii gol dberg s theorem a meromorphic function on c which is the solution of a first order algebraic differential equation has finite order and iii the fatou julia theorem the julia set of a rational function of degree d 2 is the closure of the repelling periodic points we also discuss bloch s principle and provide simple solutions to some problems of hayman connected with this principle over twenty years ago on the way to a partial explication of the phenomenon known as bloch s principle i proved a little lemma characterizing normal families of holomorphic and meromorphic functions on plane domains 68 over the years the lemma has grown and in dextrous hands proved amazingly versatile with applications to a wide variety of topics in function theory and related areas with the renewed interest in normal families arising largely from the important role they play in complex dynamics it seems sensible to survey some of the most striking of these applications to the one variable theory with the aim of making this technique available to as broad an audience as possible that is the purpose of this report one pleasant aspect of the theory is that judicious application of the lemma often leads to proofs which seem almost magical in their brevity in such cases we have made no effort to resist the temptation to write out complete proofs hardly anything beyond a basic knowledge of function theory is required to understand what follows so the reader is urged to take courage and plough on through and now we turn to our tale 1 let d be a domain in the complex plane c we shall be concerned with analytic maps i e meromorphic functions f d r2 received by the editors october 15 1997 and in revised form may 26 1998 1991 mathematics subject classification primary 30d45 secondary 30d35 34a20 58f23

strong stability of bounded evolution families and semigroups abstract we prove several characterizations of strong stability of uniformly bounded evolution families u t s t s 0 of bounded operators on a banach space x i e we characterize the property lim t u t s x 0 for all s 0 and all x x these results are connected to the asymptotic stability of the well posed linear nonautonomous cauchy problem u t a t u t t s 0 u s x x x in the autonomous case i e when u t s t t s for some c 0 semigroup t t t 0 we present in addition a range condition on the generator a of t t t 0 which is sufficient for strong stability this condition is more general than the condition in the ablv theorem involving countability of the imaginary part of the spectrum of a

stability radius and internal versus external stability in banach spaces an evolution semigroup approach in this paper the theory of evolution semigroups is developed and used to provide a framework to study the stability of general linear control systems these include autonomous and nonautonomous systems modeled with unbounded state space operators acting on banach spaces this approach allows one to apply the classical theory of strongly continuous semigroups to time varying systems in particular the complex stability radius may be expressed explicitly in terms of the generator of an evolution semigroup examples are given to show that classical formulas for the stability radius of an autonomous hilbert space system fail in more general settings upper and lower bounds on the stability radius are proven for banach space systems in addition it is shown that the theory of evolution semigroups allows for a straightforward operator theoretic analysis of internal stability as determined by classical frequency domain and input output operators even for nonautonomous banach space systems in particular for the nonautonomous setting internal stability is shown to be equivalent to input output stability for stabilizable and detectable systems for the autonomous setting an explicit formula for the norm of input output operator is given

tauberian theorems and stability of one parameter semigroups the main result is the following stability theorem let t t t tzo be a bounded co semigroup on a reflexive space x denote by a the generator of t and by a a the spectrum of a if a a n ir is countable and no eigenvalue of a lies on the imaginary axis then limt boo t t x o for all x e x

normal families of meromorphic functions concerning shared values abstract in this paper we study the problem of normal families of meromorphic functions concerning shared values and prove that a family f of meromorphic functions in a domain d is normal if for each pair of functions f and g in f f a f n and g a g n share a value b in d where n is a positive integer and a b are two finite constants such that n 4 and a 0 this result is not true when n 3

monotonicity results for coherent mimo rician channels the dependence of the gaussian input information rate on the line of sight los matrix in multiple input multiple output mimo coherent rician fading channels is explored it is proved that the outage probability and the mutual information induced by a multivariate circularly symmetric gaussian input with any covariance matrix are monotonic in the los matrix d or more precisely monotonic in d sup spl dagger d in the sense of the loewner partial order conversely it is also demonstrated that this ordering on the los matrices is a necessary condition for the uniform monotonicity over all input covariance matrices this result is subsequently applied to prove the monotonicity of the isotropic gaussian input information rate and channel capacity in the singular values of the los matrix extensions to multiple access channels mac are also provided

a deterministic equivalent for the analysis of non gaussian correlated mimo multiple access channels using large dimensional random matrix theory rmt we conduct mutual information analysis of a multiple input multiple output mimo multiple access channel mac our channel model reflects the characteristics in small cell networks where antenna correlations line of sight components and general type of fading distributions have to be included the mutual information expression can be expressed as functionals of the stieltjes transform through the so called shannon transform ideally if the stieltjes transform is known in the context of the large dimensional rmt then the problem is solved however it is difficult to derive the stieltjes transform of the considered channel models directly especially when the transmit correlation matrices are generally nonnegative definite and the channel entries are non gaussian to overcome this we use the generalized lindeberg principle to show that the stieltjes transforms of this class of random matrices with gaussian or non gaussian independent entries coincide in the large dimensional regime this result permits to derive the deterministic equivalents e g the stieltjes transform and the ergodic mutual information for non gaussian mimo channels from the known results developed for gaussian mimo channels as an application we determine the capacity achieving input covariance matrices for the mimo macs and prove that the capacity achieving input covariance matrices are asymptotically independent of the fading distribution

optimality of beamforming in fading mimo multiple access channels we consider the sum capacity of a multi input multi output mimo multiple access channel mac where the receiver has the perfect channel state information csi while the transmitters have either no or partial csi when the transmitters have partial csi it is in the form of either the covariance matrix of the channel or the mean matrix of the channel for the covariance feedback case we mainly consider physical models that result in single sided correlation structures for the mean feedback case we consider physical models that result in in phase received signals under these assumptions we analyze the mimo mac from three different viewpoints first we consider a finite sized system we show that the optimum transmit directions of each user are the eigenvectors of its own channel covariance and mean feedback matrices in the covariance and mean feedback models respectively also we find the conditions under which beamforming is optimal for all users second in the covariance feedback case we prove that the region where beamforming is optimal for all users gets larger with the addition of new users into the system in the mean feedback case we show through simulations that this is not necessarily true third we consider the asymptotic case where the number of users is large we show that in both no and partial csi cases beamforming is asymptotically optimal in particular in the case of no csi we show that a simple form of beamforming which may be characterized as an arbitrary antenna selection scheme achieves the sum capacity in the case of partial csi we show that beamforming in the direction of the strongest eigenvector of the channel feedback matrix achieves the sum capacity finally we generalize our covariance feedback results to double sided correlation structures in the appendix

the operator equationax xb c admissibility and asymptotic behavior of differential equations it is shown that there exists a natural relationship between the regular admissibility of a translation invariant subspace m ofbuc r e the space of uniformly continuous bounded functions onrwith values in a banach spacee w r t the differential equationu t au t f t and the unique solvability of a special operator equation of lyapunov s typeax xdm 0 where dmis the restriction of the operator d d dtonto m and 0 f f 0 this leads to some spectral conditions for the regular admissibility applications to questions of exponential dichotomy exponential stability the existence of periodic and almost periodic solutions of equation as well as analogous questions for some nonlinear equations and functional differential equations are presented

on normal families of meromorphic functions in this paper we study the normality of a family of meromorphic functions and obtain some normality results for meromorphic functions which improve and generalize the related results of gu bergweiler and lin

theorems of katznelson tzafriri type for contractions for t a contraction on a hilbert space and a function in the disc algebra we show that if 0 on spt then limn t tn 0 the same method gives a new proof of recent results of katznelson tzafriri several applications are given

asymptotic mutual information statistics of separately correlated rician fading mimo channels the asymptotic probability distribution of the mutual information of a separately correlated rician fading multiple input multiple output mimo channel is addressed the mean and variance of the mutual information are derived when the number of transmit and receive antennas grows asymptotically large while their ratio approaches a finite constant this derivation is based on the replica method widely used in theoretical physics and more recently in the analysis of communication systems code division multiple access cdma and mimo though the replica method allows to analyze complex systems in a comparatively simple way some authors pointed out that its assumptions are not always rigorous it is shown that the mutual information converges asymptotically to a gaussian distribution under mild technical conditions which are tantamount to assuming that the spatial correlation structure has no asymptotically dominant eigenmodes the accuracy of the asymptotic approach is assessed by numerical results it is shown that the approximation is very accurate in a wide range of system settings even when the number of transmit and receive antennas is as small as a few units

evolution semigroups and spectral criteria for almost periodic solutions of periodic evolution equations abstract we investigate spectral criteria for the existence of almost periodic solutions to linear 1 periodic evolution equations of the formdx dt a t x f t with in general unbounded a t and almost periodicf using the evolution semigroup associated with the evolutionary process generated by the equation under consideration we show that if the spectrum of the monodromy operator does not intersect the set e isp f then the above equation has an almost periodic mild solutionxfwhich is unique if one requiressp xf 2 k k z sp f we emphasize that our method allows us to treat the equations without assumption on the existence of floquet representation this improves recent results on the subject in addition we discuss some particular cases in which the spectrum of monodromy operator does not intersect the unit circle and apply the obtained results to study the asymptotic behavior of solutions finally an application to parabolic equations is considered

on the spectrum complete trajectories and asymptotic stability of linear semi dynamical systems abstract let t t t 0 be a c0 semigroup in a banach space x with generator a we prove that if t t t 0 is bounded and sun reflexive and the sun dual semigroup is not asymptotically stable then there exist bounded complete trajectories under t t t 0 provided i there is t0 0 such that ran t t0 is dense in x or ii a i r questions of almost periodicity of complete trajectories are also discussed and a new proof of our earlier theorem jointly with yu i lyubich on asymptotic stability is given

on the capacity achieving covariance matrix for frequency selective mimo channels using the asymptotic approach in this contribution an algorithm for evaluating the capacity achieving input covariance matrices for frequency selective rayleigh mimo channels is proposed in contrast with the flat fading rayleigh case no closed form expressions for the eigenvectors of the optimum input covariance matrix are available classically both the eigenvectors and eigenvalues are computed numerically and the corresponding optimization algorithms remain computationally very demanding in this paper it is proposed to optimize w r t the input covariance matrix a large system approximation of the average mutual information derived by moustakas and simon the validity of this asymptotic approximation is clarified thanks to gaussian large random matrices methods it is shown that the approximation is a strictly concave function of the input covariance matrix and that the average mutual information evaluated at the argmax of the approximation is equal to the capacity of the channel up to a o 1 t term where t is the number of transmit antennas an algorithm based on an iterative waterfilling scheme is proposed to maximize the average mutual information approximation and its convergence studied numerical simulation results show that even for a moderate number of transmit and receive antennas the new approach provides the same results as direct maximization approaches of the average mutual information

on the capacity achieving covariance matrix for rician mimo channels an asymptotic approach in this paper the capacity achieving input covariance matrices for coherent block fading correlated multiple input multiple output mimo rician channels are determined in contrast with the rayleigh and uncorrelated rician cases no closed form expressions for the eigenvectors of the optimum input covariance matrix are available classically both the eigenvectors and eigenvalues are computed numerically and the corresponding optimization algorithms remain computationally very demanding in the asymptotic regime where the number of transmit and receive antennas converge to infinity at the same rate new results related to the accuracy of the approximation of the average mutual information are provided based on the accuracy of this approximation an attractive optimization algorithm is proposed and analyzed this algorithm is shown to yield an effective way to compute the capacity achieving matrix for the average mutual information and numerical simulation results show that even for a moderate number of transmit and receive antennas the new approach provides the same results as direct maximization approaches of the average mutual information

capacity bounds via duality with applications to multiple antenna systems on flat fading channels a technique is proposed for the derivation of upper bounds on channel capacity it is based on a dual expression for channel capacity where the maximization of mutual information over distributions on the channel input alphabet is replaced with a minimization of average relative entropy over distributions on the channel output alphabet we also propose a technique for the analysis of the asymptotic capacity of cost constrained channels the technique is based on the observation that under fairly mild conditions capacity achieving input distributions escape to infinity the above techniques are applied to multiple antenna flat fading channels with memory where the realization of the fading process is unknown at the transmitter and unknown or only partially known at the receiver it is demonstrated that for high signal to noise ratio snr the capacity of such channels typically grows only double logarithmically in the snr to better understand this phenomenon and the rates at which it occurs we introduce the fading number as the second order term in the high snr asymptotic expansion of capacity and derive estimates on its value for various systems it is suggested that at rates that are significantly higher than the fading number communication becomes extremely power inefficient thus posing a practical limit on practically achievable rates upper and lower bounds on the fading number are also presented for single input single output siso systems the bounds coincide thus yielding a complete characterization of the fading number for general stationary and ergodic fading processes we also demonstrate that for memoryless multiple input single output miso channels the fading number is achievable using beam forming and we derive an expression for the optimal beam direction this direction depends on the fading law and is in general not the direction that maximizes the snr on the induced siso channel using a new closed form expression for the expectation of the logarithm of a noncentral chi square distributed random variable we provide some closed form expressions for the fading number of some systems with gaussian fading including siso systems with circularly symmetric stationary and ergodic gaussian fading the fading number of the latter is determined by the fading mean fading variance and the mean squared error in predicting the present fading from its past it is not directly related to the doppler spread for the rayleigh ricean and multiple antenna rayleigh fading channels we also present firm nonasymptotic upper and lower bounds on channel capacity these bounds are asymptotically tight in the sense that their difference from capacity approaches zero at high snr and their ratio to capacity approaches one at low snr

the brunn minkowski inequality in 1978 osserman 124 wrote an extensive survey on the isoperimetric inequality the brunn minkowski inequality can be proved in a page yet quickly yields the classical isoperimetric inequality for important classes of subsets of rn and deserves to be better known this guide explains the relationship between the brunn minkowski inequality and other inequalities in geometry and analysis and some applications

almost periodicity of mild solutions of inhomogeneous periodic cauchy problems abstract we consider a mild solution u of a well posed inhomogeneous cauchy problem u t a t u t f t on a banach space x where a is periodic for a problem on r we show that u is asymptotically almost periodic if f is asymptotically almost periodic u is bounded uniformly continuous and totally ergodic and the spectrum of the monodromy operator v contains only countably many points of the unit circle for a problem on r we show that a bounded uniformly continuous solution u is almost periodic if f is almost periodic and various supplementary conditions are satisfied we also show that there is a unique bounded solution subject to certain spectral assumptions on v f and u

tauberian theorems and stability of solutions of the cauchy problem let f r x be a bounded strongly measurable function with values in a banach space x and let ie be the singular set of the laplace transform f in ir suppose that e is countable and 0 e i uf s u du 0 uniformly for s 0 as 0 for each in e it is shown that t 0 e i uf u du f i 0 as t for each in r e in particular f t 0 if f is uniformly continuous this result is similar to a tauberian theorem of arendt and batty it is obtained by applying a result of the authors concerning local stability of bounded semigroups to the translation semigroup on buc r x and it implies several results concerning stability of solutions of cauchy problems

almost periodic solutions of first and second order cauchy problems abstract almost periodicity of solutions of first and second order cauchy problems on the real line is proved under the assumption that the imaginary resp real spectrum of the underlying operator is countable related results have been obtained by ruess v and basit our proof uses a new idea it is based on a factorisation method which also gives a short proof of the vector valued version of loomis classical theorem saying that a bounded uniformly continuous function from r into a banach spacexwith countable spectrum is almost periodic ifc0 x our method can also be used for solutions on the half line this is done in a separate paper

shannon theoretic approach to a gaussian cellular multiple access channel we obtain shannon theoretic limits for a very simple cellular multiple access system in our model the received signal at a given cell site is the sum of the signals transmitted from within that cell plus a factor spl alpha 0 spl les spl alpha spl les 1 times the sum of the signals transmitted from the adjacent cells plus ambient gaussian noise although this simple model is scarcely realistic it nevertheless has enough meat so that the results yield considerable insight into the workings of real systems we consider both a one dimensional linear cellular array and the familiar two dimensional hexagonal cellular pattern the discrete time channel is memoryless we assume that n contiguous cells have active transmitters in the one dimensional case and that n sup 2 contiguous cells have active transmitters in the two dimensional case there are k transmitters per cell most of our results are obtained for the limiting case as n spl rarr spl infin the results include the following 1 we define c sub n c spl circ sub n as the largest achievable rate per transmitter in the usual shannon theoretic sense in the one and two dimensional cases respectively assuming that all signals are jointly decoded we find expressions for limn spl rarr spl infin c sub n and limn spl rarr spl infin c spl circ sub n 2 as the interference parameter spl alpha increases from 0 c sub n and c spl circ sub n increase or decrease according to whether the signal to noise ratio is less than or greater than unity 3 optimal performance is attainable using tdma within the cell but using tdma for adjacent cells is distinctly suboptimal 4 we suggest a scheme which does not require joint decoding of all the users and is in many cases close to optimal

transmit precoding in a noncoherent relay channel with channel mean feedback this paper investigates the optimal source transmit precoding to maximize the ergodic mutual information between the input and the output of a noncoherent amplify and forward relay channel with multiple antennas at the source and with a single antenna at both the relay and the destination it is assumed that only the source relay channel mean information is available at the source the challenge here is that relaying introduces a nonconvex structure in the objective function therefore previous methods dealing with channel mean feedback which generally require the concavity of the objective function cannot be applied to circumvent the difficulty at hand a different approach based on stochastic ordering is employed the stochastic optimization problem here is ultimately transformed into the comparison of two nonnegative random variables in the laplace transform order it is shown that the optimal source transmit strategy is to transmit along the known channel mean and its orthogonal eigen channels our result subsumes as an asymptotic case the optimal precoding for multiple input single output miso channels without relaying under mean feedback furthermore the analysis can be partially extended to the case with multiple antennas at the relay numerical examples are provided to complement and corroborate the analysis