transmitter optimization for the multi antenna downlink with per antenna power constraints this paper considers the transmitter optimization problem for a multiuser downlink channel with multiple transmit antennas at the base station in contrast to the conventional sum power constraint on the transmit antennas this paper adopts a more realistic per antenna power constraint because in practical implementations each antenna is equipped with its own power amplifier and is limited individually by the linearity of the amplifier assuming perfect channel knowledge at the transmitter this paper investigates two different transmission schemes under the per antenna power constraint a minimum power beamforming design for downlink channels with a single antenna at each remote user and a capacity achieving transmitter design for downlink channels with multiple antennas at each remote user it is shown that in both cases the per antenna downlink transmitter optimization problem may be transformed into a dual uplink problem with an uncertain noise this generalizes previous uplink downlink duality results and transforms the per antenna transmitter optimization problem into an equivalent minimax optimization problem further it is shown that various notions of uplink downlink duality may be unified under a lagrangian duality framework this new interpretation of duality gives rise to efficient numerical optimization techniques for solving the downlink per antenna transmitter optimization problem

transfer functions of regular linear systems part ii the system operator and the lax phillips semigroup this paper is a sequel to a paper by the second author on regular linear systems 1994 referred to here as part i we introduce the system operator of a well posed linear system which for a finite dimensional system described by x ax bu y cx du would be the s dependent matrix s s a si c b d in the general case s s is an unbounded operator and we show that it can be split into four blocks as in the finite dimensional case but the splitting is not unique the upper row consists of the uniquely determined blocks a si and b as in the finite dimensional case but the lower row is more problematic for weakly regular systems which are introduced and studied here there exists a special splitting of s s where the right lower block is the feedthrough operator of the system using s 0 we give representation theorems which generalize those from part i to well posed linear systems and also to the situation when the initial time is we also introduce the lax phillips semigroup t induced by a well posed linear system which is in fact an alternative representation of a system used in scattering theory our concept of a lax phillips semigroup differs in several respects from the classical one for example by allowing an index r which determines an exponential weight in the input and output spaces this index allows us to characterize the spectrum of a and also the points where s s is not invertible in terms of the spectrum of the generator of for various values of the system is dissipative if and only if with index zero is a contraction semigroup

on gaussian mimo bc mac duality with multiple transmit covariance constraints owing to the special structure of the gaussian multiple input multiple output mimo broadcast channel bc the associated capacity region computation and beamforming optimization problems are typically non convex and thus cannot be solved directly one feasible approach is to consider the respective dual multiple access channel mac problems which are easier to deal with due to their convexity properties the conventional bc mac duality has been established via bc mac signal transformation and is applicable only for the case in which the mimo bc is subject to a single transmit sum power constraint an alternative approach is based on minimax duality which can be applied to the case of the sum power constraint or per antenna power constraint in this paper the conventional bc mac duality is extended to the general linear transmit covariance constraint ltcc case which includes sum power and per antenna power constraints as special cases the obtained general bc mac duality is applied to solve the capacity region computation for the mimo bc and beamforming optimization for the multiple input single output miso bc respectively with multiple ltccs the relationship between this new general bc mac duality and the minimax duality is also discussed and it is shown that the general bc mac duality leads to simpler problem formulations moreover the general bc mac duality is extended to deal with the case of nonlinear transmit covariance constraints in the mimo bc

feedbacks for nonautonomous regular linear systems we introduce nonautonomous well posed and absolutely regular linear systems as quadruples consisting of an evolution family and output input and input output maps subject to natural hypotheses in the spirit of weiss work these maps are represented in terms of admissible observation and control operators the latter in an approximate sense in the time domain in this setting the closed loop system exists for a canonical class of admissible feedbacks and it inherits the absolute regularity and other properties of the given system in particular we can iterate feedbacks

regular linear systems governed by neutral fdes the aim of this paper is to prove that a class of distributed parameter systems governed by neutral fdes provides regular linear systems employing the well established theory of representation transfer function and feedback of these later we then give new representations of the state and the output function of the neutral systems

how to get a conservative well posed linear system out of thin air part ii controllability and stability let a0 be a possibly unbounded positive operator on the hilbert space h which is boundedly invertible let c0 be a bounded operator from cal d a 0 1 2 with the norm z 1 2 2 langle

conservative boundary control systems abstract we study continuous time linear dynamical systems of boundary control observation type satisfying a green lagrange identity particular attention is paid to systems which have a well defined dynamics both in the forward and the backward time directions as we change the direction of time we also interchange inputs and outputs we show that such a boundary control observation system gives rise to a continuous time livsic brodski system node with strictly unbounded control and observation operators the converse is also true we illustrate the theory by a classical example namely the wave equation describing the reflecting mirror

well posedness of boundary controlled and observed stochastic port hamiltonian systems in this article stochastic port hamiltonian systems sphs on infinite dimensional spaces governed by ito stochastic differential equations sdes are introduced and some properties of this new class of systems are studied they are an extension of sphss defined on a finite dimensional state space the concept of well posedness in the sense of weiss and salamon is generalized to the stochastic context under this extended definition sphss are shown to be well posed the theory is illustrated on an example of a vibrating string subject to a hilbert space valued gaussian white noise process

admissibility of unbounded control operators for linear systems described by dot x t ax t bu t where a generates a semigroup on the state space x and b is an unbounded operator some necessary as well as some sufficient conditions are given for b to be admissible i e for any t the state x t should be in x and should depend continuously on the input u in l p this approach begins with an axiomatic description of such a system in terms of a functional equation the results are applied to the wave equation on a bounded interval

feedthrough operator for linear elasticity system with boundary control and observation we are concerned with the feedthrough operator for the open loop system of linear elasticity with dirichlet boundary control and collocated observation this system has been known to be regular in the sense of g weiss in this paper the analytic expression of the corresponding feedthrough operator is presented by means of differential geometry and fourier transform methods

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

survey paper well posed systems the lti case and beyond this survey is an introduction to well posed linear time invariant lti systems for non specialists we recall the more general concept of a system node classical and generalized solutions of system equations criteria for well posedness the subclass of regular linear systems some of the available linear feedback theory motivated by physical examples we recall the concepts of impedance passive and scattering passive systems conservative systems and systems with a special structure that belong to these classes we illustrate this theory by examples of systems governed by heat and wave equations we develop local and global well posedness results for lti systems with nonlinear in particular bilinear feedback by extracting the abstract idea behind various proofs in the literature we apply these abstract results to derive well posedness results for the burgers and navier stokes equations

dirac structures and boundary control systems associated with skew symmetric differential operators associated with a skew symmetric linear operator on the spatial domain a b we define a dirac structure which includes the port variables on the boundary of this spatial domain this dirac structure is a subspace of a hilbert space naturally associated with this dirac structure is an infinite dimensional system we parameterize the boundary port variables for which the c 0 semigroup associated with this system is contractive or unitary furthermore this parameterization is used to split the boundary port variables into inputs and outputs similarly we define a linear port controlled hamiltonian system associated with the previously defined dirac structure and a symmetric positive operator defining the energy of the system we illustrate this theory on the example of the timoshenko beam

linear transceiver design for interference alignment complexity and computation consider a multiple input multiple output mimo interference channel where each transmitter and receiver are equipped with multiple antennas an effective approach to practically achieving high system throughput is to deploy linear transceivers or beamformers that can optimally exploit the spatial characteristics of the channel the recent work of cadambe and jafar ieee trans inf theory vol 54 no 8 suggests that optimal beamformers should maximize the total degrees of freedom and achieve interference alignment in the high signal to noise ratio snr regime in this paper we first consider the interference alignment problem without channel extension and prove that the problem of maximizing the total achieved degrees of freedom for a given mimo interference channel is np hard furthermore we show that even checking the achievability of a given tuple of degrees of freedom for all receivers is np hard when each receiver is equipped with at least three antennas interestingly the same problem becomes polynomial time solvable when each transmit receive node is equipped with no more than two antennas we also propose a distributed algorithm for transmit covariance matrix design that does not require the dof tuple preassignment under the assumption that each receiver uses a linear minimum mean square error mmse beamformer the simulation results show that the proposed algorithm outperforms the existing interference alignment algorithms in terms of system throughput

the worst additive noise under a covariance constraint the maximum entropy noise under a lag p autocorrelation constraint is known by burg s theorem to be the pth order gauss markov process satisfying these constraints the question is what is the worst additive noise for a communication channel given these constraints is it the maximum entropy noise the problem becomes one of extremizing the mutual information over all noise processes with covariances satisfying the correlation constraints r sub 0 r sub p for high signal powers the worst additive noise is gauss markov of order p as expected but for low powers the worst additive noise is gaussian with a covariance matrix in a convex set which depends on the signal power

infinite dimensional linear systems a distributional approach we introduce the concept of a distributional resolvent linear system and solve the linear quadratic optimal control problem for this class of systems the class of distributional resolvent linear systems includes all linear time invariant systems that have been studied in the control literature we show connections with integrated semigroups and distribution semigroups and we consider some examples of systems described by partial differential equations that could not be handled by the existing theory

conditions for robustness and nonrobustness of thestability of feedback systems with respect to small delays inthe feedback loop it has been observed that for many stable feedback control systems the introduction of arbitrarily small time delays into the loop causes instability in this paper we present a systematic frequency domain treatment of this phenomenon for distributed parameter systems we consider the class of all matrix valued transfer functions which are bounded on some right half plane and which have a limit at spinfty along the real axis such transfer functions are called regular under the assumption that a regular transfer function is stabilized by unity output feedback we give sufficient conditions for the robustness and for the nonrobustness of the stability with respect to small time delays in the loop these conditions are given in terms of the high frequency behavior of the open loop system moreover we discuss robustness of stability with respect to small delays for feedback systems with dynamic compensators in particular we show that if a plant with infinitely many poles in the closed right half plane is stabilized by a controller then the stability is not robust with respect to delays we show that the instability created by small delays is itself robust to small delays three examples are given to illustrate these results

the regularity of the wave equation with partial dirichlet control and colocated observation in this paper we analyze a multidimensional controlled wave equation on a bounded domain subject to partial dirichlet control and colocated observation by means of a partial fourier transform it is shown that the system is well posed and regular in the sense of d salamon and g weiss the corresponding feedthrough operator is found to be the identity operator on the input space

uplink downlink duality via minimax duality the sum capacity of a gaussian vector broadcast channel is the saddle point of a minimax gaussian mutual information expression where the maximization is over the set of transmit covariance matrices subject to a power constraint and the minimization is over the set of noise covariance matrices subject to a diagonal constraint this sum capacity result has been proved using two different methods one based on decision feedback equalization and the other based on a duality between uplink and downlink channels this paper illustrates the connection between the two approaches by establishing that uplink downlink duality is equivalent to lagrangian duality in minimax optimization this minimax lagrangian duality relation allows the optimal transmit covariance and the least favorable noise covariance matrices in a gaussian vector broadcast channel to be characterized in terms of the dual variables in particular it reveals that the least favorable noise is not unique further the new lagrangian interpretation of uplink downlink duality allows the duality relation to be generalized to gaussian vector broadcast channels with arbitrary linear constraints however duality depends critically on the linearity of input constraints duality breaks down when the input constraint is an arbitrary convex constraint this shows that the minimax representation of the broadcast channel sum capacity is more general than the uplink downlink duality representation

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

packing of regular tetrahedral quartets of circles on a sphere how must 4n nonoverlapping equal circles forming n quartets be packed on a sphere so that the angular diameter of the circles will be as large as possible under the constraint that within each qua

the state feedback regulator problem for regular linear systems this paper is about the state feedback regulator problem for infinite dimensional linear systems the plant assumed to be an exponentially stable regular linear system is driven by a linear possibly infinite dimensional exosystem via a disturbance signal the exosystem has its spectrum in the closed right half plane and also generates the reference signal for the plant output the regulator problem is to design a controller that while guaranteeing the stability of the closed loop system without the exosystem drives the tracking error to zero a particular version of this problem is the state feedback regulator problem in which the states of the exosystem and the plant are known to the controller under suitable assumptions we show that the latter problem is solvable if and only if a pair of algebraic equations called the regulator equations is solvable we derive conditions in terms of the transfer function of the plant and eigenvalues of the exosystem for the solvability of the regulator equations three examples illustrating the theory are presented

controllability and stability of a second order hyperbolic system with collocated sensor actuator a second order hyperbolic system with collocated sensor actuator is considered the semigroup generation is shown for the closed loop system under the feedback of a generic unbounded observation operator the equivalence between the exponential stability of the closed loop system and exact controllability of the open loop system is established in the general framework of well posed linear systems finally the conditions are weakened for the diagonal semigroups with finite dimensional inputs example of beam equation is presented to display the application

achievable rates of multi antenna downlink channels with peak power constraints this paper considers a gaussian multiple input single output miso broadcast channel with a per antenna peak power constraint or simply peak power constraint it is more realistic to consider the peak power constraint on each transmit antenna because in many practical implementations each antenna is equipped with its own power amplifier assuming the perfect knowledge of the channel state information csi at the transmitter we propose an achievable scheme using dirty tape coding dtc the ideal dirty paper coding dpc which is a capacity achieving scheme for the gaussian multiple input multiple output mimo broadcast channel cannot be used for our model because its optimal input distribution is gaussian and thus the peak power constraint is violated on the other hand the channel input of dtc is uniformly distributed in a fixed range which helps to control the peak power of the transmit signal easily we also present an algorithm that finds capacity achieving beamforming vectors and power allocation factors under a per antenna average power constraint and use the optimized parameters in the proposed scheme simulation results show that the proposed scheme provides gains of 2 7db over a non dtc scheme based on minimum mean square error mmse beamforming at high signal to noise ratio snr when there are three receivers and the transmitter has three antennas

mimo b mac interference network optimization under rate constraints by polite water filling and duality we take two new approaches to design efficient algorithms for transmitter optimization under rate constraints in order to guarantee the quality of service for mimo b mac interference networks a b mac network is a generalized interference network that is a combination of multiple interfering broadcast channels bc and multiaccess channels mac two related optimization problems maximizing the minimum of weighted rates under a sum power constraint and minimizing the sum power under rate constraints are considered the first approach takes advantage of existing algorithms for sinr problems by building a bridge between rate and sinr through the design of optimal mappings between them the second approach exploits the polite water filling structure which is the network version of water filling satisfied by all the pareto optimal input of a large class of achievable regions of b mac networks it replaces most generic optimization algorithms currently used for such networks and reduces the complexity while demonstrating superior performance even in non convex cases both centralized and distributed algorithms are designed and the performance is analyzed in addition to numeric examples

secure transmission with multiple antennas part ii the mimome wiretap channel the capacity of the gaussian wiretap channel model is analyzed when there are multiple antennas at the sender intended receiver and eavesdropper the associated channel matrices are fixed and known to all the terminals a computable characterization of the secrecy capacity is established as the saddle point solution to a minimax problem the converse is based on a sato type argument used in other broadcast settings and the coding theorem is based on gaussian wiretap codebooks at high signal to noise ratio snr the secrecy capacity is shown to be attained by simultaneously diagonalizing the channel matrices via the generalized singular value decomposition and independently coding across the resulting parallel channels the associated capacity is expressed in terms of the corresponding generalized singular values it is shown that a semi blind masked multi input multi output mimo transmission strategy that sends information along directions in which there is gain to the intended receiver and synthetic noise along directions in which there is not can be arbitrarily far from capacity in this regime necessary and sufficient conditions for the secrecy capacity to be zero are provided which simplify in the limit of many antennas when the entries of the channel matrices are independent and identically distributed the resulting scaling laws establish that to prevent secure communication the eavesdropper needs three times as many antennas as the sender and intended receiver have jointly and that the optimum division of antennas between sender and intended receiver is in the ratio of 2 1

dynamic stabilization of regular linear systems we consider a general class of infinite dimensional linear systems called regular linear systems for which convenient representations are known to exist both in time and in frequency domain for this class of systems we investigate the concepts of stabilizability and detectability in particular their invariance under feedback and their relationship to exponential stability we introduce two concepts of dynamic stabilization the first formulated as usual with the plant and the controller connected in feedback and the second with two feedback loops even for finite dimensional systems the second concept stabilization with an internal loop in the controller is more general we argue that the more general concept is the natural one and we derive sufficient conditions under which an observer based stabilizing controller with an internal loop can be constructed

boundary and distributed control of the viscous burgers equation in this paper the dynamics of the forced burgers equation ut vuxx uux f x subject to both neumann boundary conditions and periodic boundary conditions using boundary and distributed control is analyzed for the boundary control problem we show that the controlled unforced burgers equation i e the closed loop system is exponentially stable when the viscosity v is known and globally asymptotically stable when v is unknown as for the distributed control problem we apply karhunen loeve decomposition on the dynamics of the forced burgers equation to generate a low dimensional dynamical system whose dynamics is similar to that of burgers equation then a feedback linearization control is used on the reduced system to exponentially stabilize the dynamics of the equation numerical simulations for the boundary and distributed controls are presented to support the analytical results

precoding for the multiantenna downlink multiuser snr gap and optimal user ordering this paper develops a practical design method for implementing tomlinson harashima precoding thp in a downlink channel with multiple antennas at the transmitter and a single antenna at each receiver a two step design process is proposed for minimizing the total transmit power while satisfying every user s minimum data rate and maximum bit error rate ber requirements first the ber and rate requirements are converted to virtual rate requirements which account for the gap to capacity introduced by practical quadrature amplitude modulation qam and thp the second step is to determine the transmit covariance matrices which specify the entire thp system that will provide these virtual rates at the minimum total transmit power as one of the main features in the proposed scheme an algorithm for finding the optimal user encoding or presubtraction order in polynomial time is proposed in addition we also propose an algorithm that finds a near optimal order but which is much less complex the proposed method outperforms existing zero forcing based thp systems in term of power efficiency

packing regular triplets of circles on a sphere this article examines how 3 n non overlapping equal circles forming n triplets must be packed on a sphere to ensure that the angular diameter of the circles will be as large as possible furthermore this problem is solved under the constraint that within each triplet the circle centres lie at the vertices of an equilateral triangle inscribed into a great circle of the sphere computer generated solutions to this optimization problem are presented for n 2 7 triplets the configurations obtained are characterized as compounds nolids of equilateral triangles

well posedness and regularity of naghdi s shell equation under boundary control and observation abstract a system of naghdi s shell equation with dirichlet boundary control and collocated observation is considered results on the associated nonhomogeneous boundary value problem are presented based on these results it is shown that the system is well posed in the sense of d salamon and regular in the sense of g weiss the expression of the corresponding feedthrough operator is explicitly found by means of riemannian geometric method and partial fourier transform these properties make this partial differential control system parallel in many ways finite dimensional ones in the general framework of well posed and regular infinite dimensional systems

the capacity region of a product of two unmatched physically degraded gaussian broadcast channels with three individual messages and a common message this paper considers a gaussian broadcast channel with two unmatched degraded components three individual messages and a common message that is intended for all three receivers it is shown that for this channel superposition coding with gaussian signalling is sufficient to achieve every point in the capacity region

maximizing sum rate and minimizing mse on multiuser downlink optimality fast algorithms and equivalence via max min sinr maximizing the minimum weighted signal to interference and noise ratio sinr minimizing the weighted sum mean square error mse and maximizing the weighted sum rate in a multiuser downlink system are three important performance objectives in nonconvex joint transceiver and power optimization where all the users have a total power constraint we show that through connections with the nonlinear perron frobenius theory jointly optimizing power and beamformers in the max min weighted sinr problem can be solved optimally in a distributed fashion then connecting these three performance objectives through the arithmetic geometric mean inequality and nonnegative matrix theory we solve the weighted sum mse minimization and the weighted sum rate maximization in the weak interference regimes using fast algorithms in the general case we first establish optimality conditions to the weighted sum mse minimization and the weighted sum rate maximization problems and provide their further connection to the max min weighted sinr problem we then propose a distributed weighted proportional sinr algorithm that leverages our fast max min weighted sinr algorithm to solve for local optimal solution of the two nonconvex problems and give conditions under which global optimality is achieved numerical results are provided to complement the analysis

weak admissibility does not imply admissibility for analytic semigroups two conjectures on admissible control operators by george weiss are disproved in this paper one conjecture says that an operator b defined on an infinite dimensional hilbert space u is an admissible control operator if for every element u in u the vector bu defines an admissible control operator the other conjecture says that b is an admissible control operator if a certain resolvent estimate is satisfied the examples given in this paper show that even for analytic semigroups the conjectures do not hold in the last section we construct a semigroup example showing that the first estimate in the hille yosida theorem is not sufficient to conclude boundedness of the semigroup

a symmetry treatment of danzerian rigidity for circle packing a symmetry extended mobility criterion that incorporates danzer s concept of bar and joint assemblies is derived and applied to the spherical circle packing problem the known scalar counting rule

transfer functions of regular linear systems part i characterizations of regularity we recall the main facts about the representation of regular linear systems essentially that they can be described by equations of the form x t ax t bu t y t cx t du t like finite dimensional systems but now a b and c are in general unbounded operators regular linear systems are a subclass of abstract linear systems we define transfer functions of abstract linear systems via a generalization of a theorem of foures and segal we prove a formula for the transfer function of a regular linear system which is similar to the formula in finite dimensions the main result is a simple to state but hard to prove necessary and sufficient condition for an abstract linear system to be regular in terms of its transfer function other conditions equivalent to regularity are also obtained the main result is a consequence of a new tauberian theorem which is of independent interest

coprime factorization for regular linear systems mild sufficient conditions are given for the existence of a doubly coprime factorization of the transfer function of a regular linear system as well as formulae for such a factorization the results are illustrated by two examples of delay systems one of which has infinitely many unstable poles

justification of input output methods for systems with unbounded control and observation the authors present definitions of feedback stabilizability and detectability that are weaker than those previously appearing in the literature stabilizability and detectability in this sense are sufficient to ensure an equivalence between internal and external stability for well posed systems furthermore stabilizability and detectability are shown to be preserved under feedback and other connections these results are linked to present a justification of input output controller design methods for systems with unbounded control and observation

optimal control of systems with a unitary semigroup and with colocated control and observation abstract we solve the quadratic optimal control problem on an infinite time interval for a class of linear systems whose state space is a hilbert space and whose operator semigroup is unitary the difficulty is that the systems in this class having unbounded control and observation operators may be ill posed we show that there is a surprisingly simple solution to the problem the optimal feedback turns out to be output feedback our approach is to use a change of variables which transforms the system into a one which according to recent research is known to be conservative we show that under a mild assumption the transfer function of this conservative system is inner and then it follows that the optimal control of this conservative system is trivial we give an example with the wave equation on an n dimensional domain with neumann control and dirichlet observation of the velocity

network duality for multiuser mimo beamforming networks and applications for any multiple input multiple output mimo network with linear beamformers it is shown that there exists a dual network that attains the same signal to interference plus noise ratio sinr performance this network duality concept is a generalization of the virtual uplink concept investigated by rashid farrokhi we first develop the dual relation using a generic duality theory in linear programming this approach naturally leads to the construction of a dual network and provides machinery for handling a generalized cost function we then consider the joint mimo beamforming and power control problem with individual sinr constraints we apply the network duality to this problem and propose a high performance iterative algorithm which compared with past centralized approaches has improved convergence behavior finally we propose a simpler distributed version of the algorithm tailored for a cellular downlink this algorithm can be readily implemented in a distributed fashion since it does not require information exchange between base stations the algorithm achieves performance close to that of centralized ones and outperforms other decentralized approaches available in the literature

transition from spherical circle packing to covering geometrical analogues of chemical isomerization how must n equal circles spherical caps of given angular radius r be arranged on the surface of a sphere so that the area covered by the circles will be as large as possible in this paper conjectured solutions of this problem for n 5 7 8 9 10 11 are given when r varies from the maximum packing radius to the minimum covering radius the relation of the results of this problem to the extremal configurations of n mutually repulsive equal point charges on the surface of a sphere is discussed variation of the solution polyhedra with r gives an abstract model of chemical isomerization processes for example for n 5 where the packing covering transition models the berry pseudo rotation of a trigonal bipyramid

sum capacity of gaussian vector broadcast channels this paper characterizes the sum capacity of a class of potentially nondegraded gaussian vector broadcast channels where a single transmitter with multiple transmit terminals sends independent information to multiple receivers coordination is allowed among the transmit terminals but not among the receive terminals the sum capacity is shown to be a saddle point of a gaussian mutual information game where a signal player chooses a transmit covariance matrix to maximize the mutual information and a fictitious noise player chooses a noise correlation to minimize the mutual information the sum capacity is achieved using a precoding strategy for gaussian channels with additive side information noncausally known at the transmitter the optimal precoding structure is shown to correspond to a decision feedback equalizer that decomposes the broadcast channel into a series of single user channels with interference pre subtracted at the transmitter

writing on dirty paper corresp a channel with output y x s z is examined the state s sim n 0 qi and the noise z sim n 0 ni are multivariate gaussian random variables i is the identity matrix the input x in r n satisfies the power constraint l n sum i 1 n x i 2 leq p if s is unknown to both transmitter and receiver then the capacity is frac 1 2 ln 1 p n q nats per channel use however if the state s is known to the encoder the capacity is shown to be c ast frac 1 2 ln 1 p n independent of q this is also the capacity of a standard gaussian channel with signal to noise power ratio p n therefore the state s does not affect the capacity of the channel even though s is unknown to the receiver it is shown that the optimal transmitter adapts its signal to the state s rather than attempting to cancel it

well posedness and exact controllability of fourth order schrodinger equation with boundary control and collocated observation in this paper we study the well posedness and exact controllability of a system described by a fourth order schrodinger equation on a bounded domain of mathbb r n n geqslant 2 with boundary control and collocated observation the neumann boundary control problem is first discussed it is shown that the system is well posed in the sense of d salamon this implies the exponential stability of the closed loop system under proportional output feedback control the well posedness result is then generalized to the dirichlet boundary control problem in particular in order to conclude feedback stabilization from well posedness we discuss the exact controllability with the dirichlet boundary control which is similar to the neumann boundary control case in addition we show that both systems are regular in the sense of g weiss and their feedthrough operators are zero

joint design of tx rx beamformers in mimo downlink channel we consider a single cell multiple input multiple output mimo downlink channel where linear transmission and reception strategy is employed the base station bs transmitter is equipped with a scheduler using a simple opportunistic beamforming strategy which associates an intended user for each of the transmitted data streams for the case when the channel of the scheduled users is available at the bs we propose a general method for joint design of the transmit and the receive beamformers according to different optimization criteria including weighted sum rate maximization weighted sum mean square error minimization minimum signal to interference plus noise ratio sinr maximization and sum power minimization under a minimum sinr constraint the proposed method can handle multiple antennas at the bs and at the mobile user with single and or multiple data streams per scheduled user the optimization problems encountered in the beamformer design e g covariance rank constraint are not convex in general therefore the problem of finding the global optimum is intrinsically nontractable however by exploiting the uplink downlink sinr duality we decompose the original optimization problem as a series of simpler optimization problems which can be efficiently solved by using standard convex optimization tools even though each subproblem is optimally solved there is no guarantee that the global optimum has been found due to the nonconvexity of the problem however the simulations show that the algorithms converge fast to a solution which can be a local optimum but is still efficient

time varying regular bilinear systems the theory of infinite dimensional systems introduced and developed by salamon trans amer math soc 300 1987 pp 383 431 and weiss siam j control optim 27 1989 pp 527 545 has been recently applied for various systems see e g the jacob partington survey current trends in operator theory and its applications birkhauser boston 2004 in the same spirit schnaubelt in his recent work siam j control optim 41 2002 pp 1141 1165 has extended their approach to the time varying linear systems in this paper we follow this spirit and prove that one can extend the results recently obtained for infinite dimensional time invariant bilinear systems to the time varying setting in particular we show that the absolute regularity and detectability assumptions ensure the existence of an observer for this kind of systems

packing of twinned circles on a sphere the twinned circle problem is to pack 2 n non overlapping equal circles forming n pairs of twins rigidly connected neighbours on a sphere so that the angular radius of the circles will be as large as possible in the case that the contact graph s of the unconstrained circle packing support s at least one perfect matching a complete solution to the twinned circles problem is found with the same angular radius as the unconstrained problem solutions for n 2 12 pairs of twins are counted and classified by symmetry for n 2 6 and 12 these are mathematically proven to be the best solutions for n 7 11 they are based on the best known conjectured solutions of the unconstrained problem where the contact graph of the unconstrained problem has one or more rattling circles the twinned problem is most easily solved by finding perfect matchings of an augmented graph in which each rattling circle is supposed to be simultaneously in contact with all its contactable neighbours the underlying contact graphs for the unconstrained packings for n 2 12 are all hamiltonian guaranteeing the existence of perfect matchings but hamiltonicity is not a necessary condition the first solution to the twins problem based on an example of a non hamiltonian contact graph occurs at n 16

on the sum rate of the gaussian mimo z channel and the gaussian mimo x channel in this paper we study the gaussian mimo z channel and the gaussian mimo x channel the mimo x channel xc consists of two multiple antenna transmit receive pairs where each transmitter communicates with both receivers the mimo z channel zc is obtained from the mimo x channel by eliminating one of the links and its corresponding message first we derive a sum rate upper bound for the mimo z channel and compare it with an existing bound in literature next we consider the mimo x channel and propose a new sum rate upper bound by utilizing the sum rate upper bound for the mimo zc subsequently we derive another upper bound for the mimo xc by assuming receiver cooperation and deriving the worst noise covariance matrix for the resulting two user mac we compare the above two upper bounds for the mimo xc with the maddahali motahari khandani mmk scheme then we consider some consequences of the above results for the mimo interference channel finally we present some numerical results the numerical results suggest that the proposed sum rate capacity upper bounds are tighter than existing bounds

counterexamples concerning observation operators for c0 semigroups this paper concerns systems of the form dot x t ax t y t cx t where a generates a c 0 semigroup two conjectures which were posed in 1991 and 1994 are shown not to hold the first conjecture by g weiss states that if the range of c is one dimensional then c is admissible if and only if a certain resolvent estimate holds the second conjecture by d russell and g weiss states that a system is exactly observable if and only if a test similar to the hautus test for finite dimensional systems holds the c 0 semigroup in both counterexamples is analytic and possesses a basis of eigenfunctions using the a c pair from the second counterexample we construct a generator a e on a hilbert space such that si a e is uniformly left invertible but its semigroup does not have this property

from circle packing to covering on a sphere with antipodal constraints the problem treated here is how must n antipodal pairs of equal circles spherical caps of given angular radius r be arranged on the surface of a sphere so that the area covered by the circles will be as large as possible conjectured solutions of this problem for n 4 5 7 are given where r varies from the maximum antipodal packing radius to the minimum antipodal covering radius these cases exhibit sequences of symmetry and edge number transitions in the contact polyhedra which in general differ from those of the unconstrained problem where centrosymmetry is not enforced new minimum antipodal covering arrangements are conjectured for 4 5 and 7 pairs they are a hexagonal bipyramid a d 2h deltahedron and the cube octahedron compound respectively

infinite dimensional linear systems with unbounded control and observation a functional analytic approach abstract the object of these notes is to develop a unifying framework for the functional analytic representation of infinite dimensional linear systems with unbounded input and output operators on the basis of the general approach new results are derived on the wellposedness of feedback systems and on the linear quadratic control problem the implications of the theory for large classes of functional and partial differential equations are discussed in detail keywords representation of infinite dimensional systems semigroups boundary control feedback linear quadratic control

exponential stabilization of well posed systems by colocated feedback we consider well posed linear systems whose state trajectories satisfy dot x ax bu where u is the input and a is an essentially skew adjoint and dissipative operator on the hilbert space x this means that the domains of a and a are equal and a a q where q geq 0 is bounded on x the control operator b is possibly unbounded but admissible and the observation operator of the system is b such a description fits many wave and beam equations with colocated sensors and actuators and it has been shown for many particular cases that the feedback u ka y v with ka 0 stabilizes the system strongly or even exponentially here y is the output of the system and v is the new input we show by means of a counterexample that if b is sufficiently unbounded then such a feedback may be unsuitable the closed loop semigroup may even grow exponentially our counterexample is a simple regular system with feedthrough operator zero however we prove that if the original system is exactly controllable and observable and if ka is sufficiently small then the closed loop system is exponentially stable

a physically motivated class of scattering passive linear systems we introduce a class of scattering passive linear systems motivated by examples from mathematical physics the state space of the system is x h oplus e where h and e are hilbert spaces we also have a hilbert space e 0 which is dense in e with continuous embedding and e 0 is the dual of e 0 with respect to the pivot space e the input space is the same as the output space and it is denoted by u the semigroup generator has the structure a big begin smallmatrix 0 l l g frac 1 2 k k end smallmatrix big where l in cal l e 0 h and k in cal l e 0 u are such that big begin smallmatrix l k end smallmatrix big with domain e 0 is closed as an unbounded operator from e to h oplus u the operator g in cal l e 0 e 0 is such that rm re langle gw 0 w 0 rangle leq 0 for all w 0 in e 0 the observation operator is c left begin matrix 0 k end matrix right the control operator is b c and the output equatio

transmit beamforming and power control for optimizing the outage probability fairness in miso networks this paper studies the joint beamforming and power control in a multiuser multi input single output network by utilizing the only statistical channel distribution information such information consists of slowly varying covariance matrices in the beamforming network that can be employed to reduce instantaneous feedback overhead in transmission utilizing solely the statistical channel information we study how to minimize the maximum outage probability under a weighted sum power constraint that guarantees max min fairness to all users this problem is however generally hard to solve due to the nonconvexity and nonlinear coupling between beamformer and power variables first assuming a fixed beamformer set we use the nonlinear perron frobenius theory to design a decentralized algorithm with provable geometrically fast convergence rate to compute the optimal power then for the general case we examine a certainty equivalent margin counterpart with outage mapped thresholds that incorporate the statistical channel information we show that a network duality for this certainty equivalent problem can be useful to decouple the coupling between the beamformer and power variables this nonlinear perron frobenius theory motivated approach yields a feasible beamformer and power allocation that are near optimal as compared to monte carlo averaging simulations

minimum phase infinite dimensional second order systems in general better performance can be achieved with a controlled minimum phase system than a controlled nonminimum phase system we show that a wide class of second order infinite dimensional systems with either velocity or position measurements are minimum phase the results are illustrated by two examples

well posedness and controllability of a class of coupled linear systems we consider coupled systems consisting of an infinite dimensional part and a finite dimensional part connected in feedback such as for example the well known scole system a beam with a rigid body attached at one end the external world interacts with the coupled system via the finite dimensional part which receives the external input and sends out the output the infinite dimensional part is assumed to be such that it becomes well posed and strictly proper when connected in cascade with an integrator under several assumptions we derive well posedness and exact controllability results for such coupled systems the first main result concerns the case when the input signal of the finite dimensional part is the difference between the external input and the feedback signal the second main result allows a more general structure for the finite dimensional part we also prove a result for the approximate controllability of coupled systems