on the calculation of the hilbert transform from interpolated data this correspondence studies the calculation of the hilbert transform of continuous functions f with continuous conjugate f from a finite set of sampling points it shows that there exists no linear operator which approximates f arbitrary well in the uniform norm from a finite number of sampling points for all possible continuous function f with continuous conjugate f however for smooth functions such linear approximation operators exist and sufficient conditions on the smoothness of the functions are presented the correspondence also examines the robustness of the calculation of the hilbert transform from interpolated data and it gives explicit error bounds it is shown that for a large class of algorithms the error grows at least proportional to the logarithm of the number of sampling points

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

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

toward a practical scheme for binary broadcast channels with varying channel quality using dirty paper coding we consider practical schemes for binary dirty paper channels and broadcast channels bcs with two receivers and varying channel quality with the bc application in mind this paper proposes a new design for binary dirty paper coding dpc by exploiting the concept of coset binning the complexity of the system is greatly reduced when compared to the existing works some design challenges of the coset binning approach are identified and addressed the proposed binary dpc system achieves similar performance to the state of the art superposition coding based system while demonstrating significant advantages in terms of complexity and flexibility of system design for binary bcs achieving the capacity generally requires the superposition of a normal channel code and a carefully designed channel code with non uniform bit distribution the non uniform bit distribution is chosen according to the channel conditions therefore to achieve the capacity for binary bcs with varying channel quality it is necessary to use quantization codes of different rates which significantly increases the implementation complexity in this paper we also propose a broadcast scheme that generalizes the concept of binary dpc which we term soft dpc by combining soft dpc with time sharing we achieve a large percentage of the capacity for a wide range of channel quality with little complexity overhead our scheme uses only one fixed pair of codes for users 1 and 2 and a single quantization code which possesses many practical advantages over traditional time sharing and superposition coding solutions and provides strictly better performance

hall polynomials for tame type abstract in the present paper we prove that hall polynomial exists for each triple of decomposition sequences which parameterize isomorphism classes of coherent sheaves of a domestic weighted projective line x over finite fields these polynomials are then used to define the generic ringel hall algebra of x as well as its drinfeld double combining this construction with a result of cramer we show that hall polynomials exist for tame quivers which not only refines a result of hubery but also confirms a conjecture of berenstein and greenstein

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

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

spatial covariance based downlink beamforming in an sdma mobile radio system on the downlink of an sdma mobile radio system different users who share the same frequency band and time slot have to be spatially separated by beamforming in this paper we analyze an approach to jointly calculate weights in such a way that all users receive their signal with a given signal to noise and interference ratio the computations are based on spatial covariance matrices which describe the user specific channels averaged over the fast fading the presented algorithm succeeds in minimizing the array transmit power while spatially separating the users no matter which rank their spatial covariance matrices have but since it needs to solve a complex nonlinear constrained optimization problem several approximations of this algorithm are presented and compared to each other on the basis of array power complexity and convergence

a proof of marton s coding theorem for the discrete memoryless broadcast channel corresp a simple proof using random partitions and typicality is given for marton s coding theorem for broadcast channels

downlink mmse transceiver optimization for multiuser mimo systems duality and sum mse minimization we address the problem of minimum mean square error mmse transceiver design for point to multipoint transmission in multiuser multiple input multiple output mimo systems we focus on the problem of minimizing the downlink sum mse under a sum power constraint it is shown that this problem can be solved efficiently by exploiting a duality between the downlink and uplink mse feasible regions we propose two different optimization frameworks for downlink mmse transceiver design the first one solves an equivalent uplink problem then the result is transferred to the original downlink problem duality ensures that any uplink mmse scheme e g linear mmse reception or mmse decision feedback equalization dfe has a downlink counterpart we propose two globally optimum algorithms based on convex optimization the basic idea of the second framework is to perform optimization in an alternating manner by switching between the virtual uplink and downlink channels this strategy exploits that the same mse can be achieved in both links for a given choice of transmit and receive filters this iteration is proven to be convergent

triangulated categories and kac moody algebras by using the ringel hall algebra approach we find a lie algebra arising in each triangulated category with t 2 1 where t is the translation functor in particular the generic form of the lie algebras determined by the root categories the 2 period orbit categories of the derived categories of finite dimensional hereditary associative algebras gives a realization of all symmetrizable kac moody lie algebras

the bridgeland s ringel hall algebra associated to an algebra with global dimension at most two abstract for any finite dimensional associative algebra with global dimension 2 we show that there is an embedding from the twisted ringel hall algebra to the bridgeland s ringel hall algebra in particular this result is true for tilted algebras and canonical algebras

comparing lusztig s algebras and hall algebras at v 1 abstract there are two abstract approaches to positive parts of quantized enveloping algebras of kac moody algebras lusztig s axiomatic approach leading to non degenerate objects in green categories and ringel s approach via hall algebras generically and for certain values of the quantum parameter these two approaches produce isomorphic objects as shown by green this note studies a case where these objects turn out to be different for v 1 the algebra arising from lusztig s approach is shown to be super commutative for any datum i whereas hall algebras are super commutative only in a trivial case

superposition signaling in broadcast interference networks it is known that superposition signaling in gaussian interference networks is capable of improving the achievable rate region however the problem of maximizing the rate gain offered by superposition signaling is computationally prohibitive even in the simplest case of two user single input single output interference networks this paper examines superposition signaling for the general multiple input multiple output broadcast gaussian interference networks the problem of maximizing either the sum rate or the minimal user s rate under superposition signaling and dirty paper coding is solved by a computationally efficient path following procedure which requires only a convex quadratic program for each iteration but ensures convergence at least to a locally optimal solution numerical results demonstrate the substantial performance advantage of the proposed approach

linear precoding via conic optimization for fixed mimo receivers in this paper the problem of designing linear precoders for fixed multiple input multiple output mimo receivers is considered two different design criteria are considered in the first the transmitted power is minimized subject to signal to interference plus noise ratio sinr constraints in the second the worst case sinr is maximized subject to a power constraint it is shown that both problems can be solved using standard conic optimization packages in addition conditions are developed for the optimal precoder for both of these problems and two simple fixed point iterations are proposed to find the solutions that satisfy these conditions the relation to the well known uplink downlink duality in the context of joint transmit beamforming and power control is also explored the proposed precoder design is general and as a special case it solves the transmit rank one beamforming problem simulation results in a multiuser system show that the resulting precoders can significantly outperform existing linear precoders

buildings and hecke algebras abstract in this paper we establish a strong connection between buildings and hecke algebras by studying two algebras of averaging operators on buildings to each locally finite regular building we associate a natural algebra b of chamber set averaging operators and when the building is affine we also define an algebra a of vertex set averaging operators we show that for appropriately parametrised hecke algebras h and h the algebra b is isomorphic to h and the algebra a is isomorphic to the centre of h on the one hand these results give a thorough understanding of the algebras a and b on the other hand they give a nice geometric and combinatorial understanding of hecke algebras and in particular of the macdonald spherical functions and the centre of affine hecke algebras our results also produce interesting examples of association schemes and polynomial hypergroups in later work we use the results here to study random walks on affine buildings

coxeter elements and root bases abstract let g be a lie algebra of type a d e with fixed cartan subalgebra h root system r and weyl group w we show that a choice of coxeter element c w gives a root basis for g moreover using the results of kirillov and thind 2010 kt we show that this root basis gives a purely combinatorial construction of g where root vectors correspond to vertices of a certain quiver and with respect to this basis the structure constants of the lie bracket are given by paths in this construction is then related to the constructions of ringel and peng and xiao

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

a complete description of the qos feasibility region in the vector broadcast channel we characterize the complete quality of service qos feasibility region and present a simple feasibility test for given qos requirements in the gaussian vector broadcast channel while most contributions in the literature recast the qos constraints into requirements for the signal to interference and noise ratios sinrs we convert them into upper bounds for the minimum mean square errors mmses instead and test feasibility in the mmse domain our main contribution is a complete description of the feasible mmse region its closure is shown to be a polytope and we find the complete set of its bounding half spaces noniteratively after a finite number of steps the polytope can easily be converted into any other qos domain like sinr or rate however the simple geometry of the mmse domain is lost in other domains once the bounding half spaces are determined any target mmse tuple can quickly be checked for feasibility by verifying its membership to the interior of the polytope for nondegenerate channels the only relevant bounding half space is essentially the lower bound on the sum mean square error no further computations are necessary contrary to existing feasibility checks which iteratively solve eigenproblems in an alternating optimization framework for every single qos requirement to test for two particular user antenna configurations we find a noniterative closed form solution for the optimum power allocation of the signal to interference ratio sir balancing

nested linear lattice codes for structured multiterminal binning network information theory promises high gains over simple point to point communication techniques at the cost of higher complexity however lack of structured coding schemes limited the practical application of these concepts so far one of the basic elements of a network code is the binning scheme wyner 1974 1978 and other researchers proposed various forms of coset codes for efficient binning yet these schemes were applicable only for lossless source or noiseless channel network coding to extend the algebraic binning approach to lossy source or noisy channel network coding previous work proposed the idea of nested codes or more specifically nested parity check codes for the binary case and nested lattices in the continuous case these ideas connect network information theory with the rich areas of linear codes and lattice codes and have strong potential for practical applications we review these developments and explore their tight relation to concepts such as combined shaping and precoding coding for memories with defects and digital watermarking we also propose a few novel applications adhering to a unified approach

the hall polynomial revisited abstract the hall polynomial g p counts subgroups of type and cotype in a finite abelian p group of type we derive a formula for g p along the lines of t klein s original formula but much simpler which shows that its expansion in powers of p 1 has nonnegative coefficients

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

high snr analysis for mimo broadcast channels dirty paper coding versus linear precoding in this correspondence we compare the achievable throughput for the optimal strategy of dirty paper coding dpc to that achieved with suboptimal and lower complexity linear precoding techniques zero forcing and block diagonalization both strategies utilize all available spatial dimensions and therefore have the same multiplexing gain but an absolute difference in terms of throughput does exist the sum rate difference between the two strategies is analytically computed at asymptotically high snr furthermore the difference is not affected by asymmetric channel behavior when each user has a different average snr weighted sum rate maximization is also considered in the process it is shown that allocating user powers in direct proportion to user weights asymptotically maximizes weighted sum rate

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

resource allocation in multiantenna systems achieving max min fairness by optimizing a sum of inverse sir we address the problem of jointly optimizing transmit powers and linear preequalizers e g beamformers for a multiuser downlink channel all results can be transferred to the dual uplink channel the link quality of each user is determined by the signal to interference ratio sir two common strategies for distributing the system resources by joint optimization of transmit powers and beamformers are 1 maximize the minimum sir to which we refer as max min fairness and 2 minimize the sum of weighted inverse sir both strategies are generally not equivalent in this paper it is shown how the weighting factors should be chosen so that the sum optimization approach achieves optimal max min fairness an efficient iterative algorithm is proposed for solving this nonlinear optimization problem monotonicity and convergence are proved

quasi spherical metrics and applications in this paper using the idea of bartnik b2 on quasi spherical metrics we continue our study on the boundary behaviors of compact manifolds with nonnegative scalar curvature and nonempty boundary unlike the previous work st of the authors and the work of liu yau ly we only assume each boundary component has nonnegative curvature which is not identically zero we also study the case that the boundary is embedded in the quotient of the infinity of the euclidean space over a finite group the regularity of the black hole boundary condition of quasi spherical metrics is also discussed

degrees of freedom of interference channels with comp transmission and reception we study the degrees of freedom dof of the k user interference channel with coordinated multipoint comp transmission and reception each message is jointly transmitted by m t successive transmitters and is jointly received by m r successive receivers we refer to this channel as the comp channel with a transmit cooperation order of m t and receive cooperation order of m r since the channel has a total of k transmit antennas and k receive antennas the maximum possible dof is equal to k we show that the comp channel has k dof if and only if m t m r geq k 1 the key idea is that the zero forcing of the interference corresponding to the i rm th message at the decoder of the j rm th message where j ne i can be viewed as a shared responsibility between the m t transmitters carrying the i rm th message and the m r receivers decoding the j rm th message for the general case we derive an outer bound that states that the dof is bounded above by left lceil k m t m r 2 2 right rceil for the special case with only comp transmission i e m r 1 we propose a scheme that can achieve k m t 1 2 dof for all k and conjecture that the result holds true for all k in the proposed coding scheme the m t transmitters carrying each message are used to cancel the interference introduced by this message at the first m t 1 receivers thereby allowing each of these receivers to enjoy 1 dof and asymptotic interference alignment is used to align the interfering signals at each other receiver to occupy half the signal space the achievability proofs are based on the notion of algebraic independence from algebraic geometry

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

root categories and simple lie algebras abstract by using the t 2 orbit category of the derived category of a hereditary algebra which is proved to be a triangulated category too we give a complete realization of a simple lie algebra this is a global version of ringel s work for the positive parts of simple lie algebras in this realization the grothendieck group provides the cartan subalgebra the indecomposable objects correspond to the chevalley basis and the structure constants are given by the evaluations of hall polynomials

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

modulo loss reduction for vector perturbation systems in this letter we present an improved precoding technique which reduces a modulo loss in vector perturbation with low complexity instead of searching perturbation vectors in the infinite lattice the proposed scheme restricts the search range by utilizing the distribution of the perturbation vector depending on transmitted data as a result we can achieve significant complexity savings at the transmitter while providing better performance compared to the original vector perturbation

schurian sp representation finite right peak pi rings and their indecomposable socle projective modules such that soc r is an essential right ideal in r and it is isomorphic to a direct sum of finitely many copies of p called a right peak of r here f f are local rings f f is a division ring imj is an fi fj bimodule and the multiplication in r is given by fi f bimodule maps cii imj im jm satisfying the natural associativity conditions we denote by p p p the right indecomposable row ideals of r

a generic approach to qos based transceiver optimization we propose a generic framework for jointly optimizing the transmit power allocation and the adaptive receive strategies in a multiuser network with individual quality of service qos requirements the qos is assumed to be a one to one mapping of the signal to interference plus noise ratio sinr the feasibility of certain target qos depends on the mutual interference and on a given sum power constraint this coupling can be modeled by interference functions which determine how the transmit powers cause interference to the individual users we show fundamental properties like continuity and feasibility for the most general case when the functions are defined by axioms additional properties are shown for the case when the interference functions are based on a parameter dependent coupling matrix which allows to apply techniques from the theory of nonnegative matrices we derive a class of iterative algorithms which exploit the matrix structure the proposed iteration converges monotonically to the global optimum starting from the same initialization it is better than the known fixed point iteration it achieves arbitrary qos values within the qos feasible region

the mass of an asymptotically flat manifold we show that the mass of an asymptotically flat n manifold is a geometric invariant the proof is based on harmonic coordinates and to develop a suitable existence theory results about elliptic operators with rough coefficients on weighted sobolev spaces are summarised some relations between the mass scalar curvature and harmonic maps are described and the positive mass theorem for n dimensional spin manifolds is proved

drinfeld double and ringel green theory of hall algebras abstract as a continuation of the work of ringel and green on hall algebras the hopf algebra structure of a extended twisted hall algebra is presented here by introducing the ringel pairing of hall algebras the drinfeld double of a hall algebra is constructed the reduced form of this drinfeld double is applicable to the generic composition algebra which gives a complete realization of quantum group this ringel green theory makes it possible to give the formulae for the computation of commutators in quantum groups in terms of hall algebras

positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature in this paper we study the boundary behaviors of compact manifolds with nonnegative scalar curvature and nonempty boundary using a general version of positive mass theorem of schoen yau and witten we prove the following theorem for any compact manifold with boundary and nonnegative scalar curvature if it is spin and its boundary can be isometrically embedded into euclidean space as a strictly convex hypersurface then the integral of mean curvature of the boundary of the manifold cannot be greater than the integral of mean curvature of the embedded image as a hypersurface in euclidean space moreover equality holds if and only if the manifold is isometric with a domain in the euclidean space conversely under the assumption that the theorem is true then one can prove the adm mass of an asymptotically flat manifold is nonnegative which is part of the positive mass theorem

optimizing zero forcing precoders for mimo broadcast systems in this paper we develop optimization techniques for linear zero forcing zf multiple input multiple output mimo broadcast bc precoders a computationally efficient technique based on qr update algorithms for implementing this optimization is then proposed the application of the proposed zf precoders and decoders to systems with both large and small numbers of users is considered the proposed optimization technique is also shown to perform very well in conjunction with nonlinear tomlinson harashima precoding both coded and uncoded performances are considered

the hall polynomials for tame quiver algebras abstract let a be a finite dimensional tame quiver algebra over a finite field k we prove that hall polynomials exist for a

a partial order on littlewood richardson tableaux abstract for a nilpotent linear transformation f v v of a vector space v of jordan type 1 l i e the sizes of the jordan blocks equal to the conjugate of let x d be the set of f stable subspaces of dimension d x d is partitioned indexed by littlewood richardson tableaux lr tableau t of shapes s with d each of which corresponds to the set s v t consisting of w x d such that the dimension of the vector space f r 1 v f t 1 w f r v f t 1 w f r 1 v f t w is equal to the number of cells squares in the rth row of t filled with the letter t for all r t 1 we define a partial order on the set lr d of lr tableaux of shapes s with d by t t if s v t is contained in the closure of s v t in the grassmaniann g v d we give a certain sufficient condition corollary c and a necessary condition corollary d for t t for this we introduce the notions of generic vectors the generic subspace w t the rational function field f t and the lr tableau t associated to a tableau t and then show the map s v t s v t let 1 w f w where t let 1 is the subtableau of an lr tableau t consisting of the cells filled with the letters greater than 1 is a fiber bundle with fiber isomorphic to a union of schubert cells and s v t is nonsingular theorem a the generic subspace w t of a tableau t is a generic point of s v t whose function field isomorphic to the rational function field f t associated to the lr tableau t theorem b

asymptotic analysis of the amount of csi feedback in mimo broadcast channels in this paper we consider a downlink communication system in which a base station bs equipped with m antennas and power constraint p communicates with n users each equipped with k receive antennas it is assumed that the users have perfect channel state information csi of their own channels while the bs only knows the partial csi provided by the receivers via a feedback channel we study the fundamental limits on the amount of feedback required at the bs to achieve the sum rate capacity of the system when bs has perfect csi for all users in the asymptotic case of n considering various signal to noise ratio snr regimes the main results of this paper can be expressed as follows 1 in the fixed snr regime where the snr does not scale with n and low snr regime where the snr is much smaller than 1 in n to achieve the 1 e portion of the sum rate capacity the total amount of feedback should scale at least with e 1 in the fixed snr regime to reduce the gap between the achievable sum rate and the sum rate capacity of the system which is defined as the sum rate gap to zero the amount of feedback should scale at least logarithmically with the sum rate capacity which is achievable by using the random beam forming rbf scheme proposed by sharif and hassibi in the low snr regime we propose an opportunistic beam forming obf scheme which is shown to be asymptotically feedback optimal 2 in the high snr regime where the snr grows to infinity as n the total amount of feedback depends on the number of receive antennas in particular to reduce the sum rate gap to zero in the case of k 1 m 1 should scale at least logarithmically with the snr in the case of k m the amount of feedback does not need to scale with the snr moreover we show that rbf is asymptotically feedback optimal in the high snr regime

a unified successive pseudoconvex approximation framework in this paper we propose a successive pseudoconvex approximation algorithm to efficiently compute stationary points for a large class of possibly nonconvex optimization problems the stationary points are obtained by solving a sequence of successively refined approximate problems each of which is much easier to solve than the original problem to achieve convergence the approximate problem only needs to exhibit a weak form of convexity namely pseudoconvexity we show that the proposed framework not only includes as special cases a number of existing methods for example the gradient method and the jacobi algorithm but also leads to new algorithms that enjoy easier implementation and faster convergence speed we also propose a novel line search method for nondifferentiable optimization problems which is carried out over a properly constructed differentiable function with the benefit of a simplified implementation as compared to the state of the art line search techniques that directly operate on the original nondifferentiable objective function the advantages of the proposed algorithm are shown both theoretically and numerically by several example applications namely multi input multi output mimo broadcast channel capacity computation energy efficiency maximization in massive mimo systems and ell 1 regularized least square problems in sparse signal recovery

degrees of freedom region of the mimo x channel we provide achievability as well as converse results for the degrees of freedom region of a multiple input multiple output mimo x channel i e a system with two transmitters two receivers each equipped with multiple antennas where independent messages need to be conveyed over fixed channels from each transmitter to each receiver the inner and outer bounds on the degrees of freedom region are tight whenever integer degrees of freedom are optimal for each message with m 1 antennas at each node we find that the total sum rate degrees of freedom are bounded above and below as 1 les eta x les 4 3 if m 1 and channel matrices are nondegenerate then the precise degrees of freedom eta x 4 3 m thus the mimo x channel has noninteger degrees of freedom when m is not a multiple of 3 simple zero forcing without dirty paper encoding or successive decoding suffices to achieve the 4 3 m degrees of freedom if the channels vary with time frequency then the channel with single antennas m 1 at all nodes has exactly 4 3 degrees of freedom the key idea for the achievability of the degrees of freedom is interference alignment i e signal spaces are aligned at receivers where they constitute interference while they are separable at receivers where they are desired we also explore the increase in degrees of freedom when some of the messages are made available to a transmitter or receiver in the manner of cognitive radio

robust minimum variance beamforming this paper introduces an extension of minimum variance beamforming that explicitly takes into account variation or uncertainty in the array response sources of this uncertainty include imprecise knowledge of the angle of arrival and uncertainty in the array manifold in our method uncertainty in the array manifold is explicitly modeled via an ellipsoid that gives the possible values of the array for a particular look direction we choose weights that minimize the total weighted power output of the array subject to the constraint that the gain should exceed unity for all array responses in this ellipsoid the robust weight selection process can be cast as a second order cone program that can be solved efficiently using lagrange multiplier techniques if the ellipsoid reduces to a single point the method coincides with capon s method we describe in detail several methods that can be used to derive an appropriate uncertainty ellipsoid for the array response we form separate uncertainty ellipsoids for each component in the signal path e g antenna electronics and then determine an aggregate uncertainty ellipsoid from these we give new results for modeling the element wise products of ellipsoids we demonstrate the robust beamforming and the ellipsoidal modeling methods with several numerical examples

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

elliptic lie algebras and tubular algebras abstract this article is to study relations between tubular algebras of ringel and elliptic lie algebras in the sense of saito yoshii using the explicit structure of the derived categories of tubular algebras given by happel ringel we prove that the elliptic lie algebra of type d 4 1 1 e 6 1 1 e 7 1 1 or e 8 1 1 is isomorphic to the ringel hall lie algebra of the root category of the tubular algebra with the same type as a by product of our proof we obtain a chevalley basis of the elliptic lie algebra following indecomposable objects of the root category of the corresponding tubular algebra this can be viewed as an analogue of the frenkel malkin vybornov theorem in which they described a chevalley basis for each untwisted affine kac moody lie algebra by using indecomposable representations of the corresponding affine quiver

on lie algebras associated with representation finite algebras abstract let be a representation finite c algebra which has hall polynomials with the universal cover which is a locally bounded directed c algebra in this paper we prove that the z lie algebra l associated with which is defined by riedtmann in 17 and the z lie algebra k associated with which is defined by ringel in 19 are isomorphic as an application we show that if is a representation finite generalized cluster tilted algebra or representation finite trivial extension algebra then k l

asymptotic stability and decay rates of homogeneous positive systems with bounded and unbounded delays there are several results on the stability of nonlinear positive systems in the presence of time delays however most of them assume that the delays are constant this paper considers time varying possibly unbounded delays and establishes asymptotic stability and bounds the decay rate of a significant class of nonlinear positive systems which includes positive linear systems as a special case specifically we present a necessary and sufficient condition for delay independent stability of continuous time positive systems whose vector fields are cooperative and homogeneous we show that global asymptotic stability of such systems is independent of the magnitude and variation of the time delays for various classes of time delays we are able to derive explicit expressions that quantify the decay rates of positive systems we also provide the corresponding counterparts for discrete time positive systems whose vector fields are nondecreasing and homogeneous

on derived hall numbers for tame quivers abstract in the present paper we study the derived hall algebra for the bounded derived category of the finite dimensional nilpotent representations of a dynkin or tame quiver over a finite field we show that for any three given objects in the bounded derived category the associated derived hall numbers are given by a rational function in the cardinalities of ground fields

distance regularity in buildings and structure constants in hecke algebras in this paper we define generalised spheres in buildings using the simplicial structure and weyl distance in the building and we derive an explicit formula for the cardinality of these spheres we prove a generalised notion of distance regularity in buildings and develop a combinatorial formula for the cardinalities of intersections of generalised spheres motivated by the classical study of algebras associated to distance regular graphs we investigate the algebras and modules of hecke operators arising from our generalised distance regularity and prove isomorphisms between these algebras and more well known parabolic hecke algebras we conclude with applications of our main results to non negativity of structure constants in parabolic hecke algebras commutativity of algebras of hecke operators double coset combinatorics in groups with bn pairs and random walks on the simplices of buildings

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

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

a realization of elliptic lie algebras of type f 4 2 2 by the ringel hall approach abstract this article is to study relations between an elliptic lie algebra g of type f 4 2 2 and the f fixed point algebra a f of a tubular algebra a of type t 3 3 3 under a frobenius morphism f using the explicit structure of the root category of the f fixed point algebra a f we prove that the elliptic lie algebra g of type f 4 2 2 is isomorphic to the ringel hall lie algebra of the root category of the f fixed point algebra a f

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

mmse decision feedback equalizers and coding i equalization results the minimum mean squared error decision feedback equalizer mmse dfe has properties that suggest that it is a canonical equalization structure for systems that combine equalization with coded modulation the structure and performance of the mmse dfe are succinctly derived using linear estimation theoretic principles in this first part of this two part paper the front end of the mmse dfe called the mean square whitened matched filter ms wmf is preferable in some ways to a matched filter or a whitened matched filter as a canonical receiver front end in a coded system the feedback filter of the mmse dfe may be implemented in the transmitter using precoding the mmse dfe can perform significantly better than a zero forcing decision feedback equalizer particularly at moderate to low snr s and on severe isi channels the mmse dfe is biased the optimum unbiased mmse dfe is the mmse dfe with the bias removed removing bias improves error probability but reduces the snr to snr sub mmse dfe u snr sub mmse dfe 1 it is shown that this snr relationship is a particular case of a very general result and that snr sub mmse dfe u gives a more realistic estimate of snr the results are extended to partial response equalization and to equalization with correlated inputs in an appendix

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

extensions of fast lipschitz optimization the need of fast distributed solvers for optimization problems in networked systems has motivated the recent development of the fast lipschitz optimization framework in such an optimization problems satisfying certain qualifying conditions such as monotonicity of the objective function and contractivity of the constraints have a unique optimal solution obtained via fast distributed algorithms that compute the fixed point of the constraints this paper extends the set of problems for which the fast lipschitz framework applies existing assumptions on the problem form are relaxed and new and generalized qualifying conditions are established by novel results based on lagrangian duality it is shown for which cases of more constraints than decision variables and less constraints than decision variables fast lipschitz optimization applies new results are obtained by imposing non strict monotonicity of the objective functions the extended fast lipschitz framework is illustrated by a number of examples including network optimization and optimal control problems

optimality of radio power control via fast lipschitz optimization in wireless network resource allocation the radio power control problems are often solved by fixed point algorithms although these algorithms give feasible problem solutions such solutions often lack notion of problem optimality this paper reconsiders well known fixed point algorithms such as those with standard and type ii standard interference functions and investigates the conditions under which they give optimal solutions the optimality is established by the recently proposed fast lipschitz optimization framework to apply such a framework the analysis is performed by a logarithmic transformation of variables that gives tractable fast lipschitz problems it is shown how the logarithmic problem constraints are contractive by the standard or type ii standard assumptions on the power control problem and how sets of cost functions fulfill the fast lipschitz qualifying conditions the analysis on nonmonotonic interference function allows establishing a new qualifying condition for fast lipschitz optimization the results are illustrated by considering power control problems with standard interference function problems with type ii standard interference functions and a case of subhomogeneous power control problems given the generality of fast lipschitz optimization compared to traditional methods for resource allocation it is concluded that such an optimization may help to determine the optimality of many resource allocation problems in wireless networks

vector coding for partial response channels a linear technique for combining equalization and coset codes on partial response channels with additive white gaussian noise is developed the technique vector coding uses a set of transmit filters or vectors to partition the channel into an independent set of parallel intersymbol interference isi free channels for any given finite or infinite block length the optimal transmit vectors for such channel partitioning are shown to be the eigenvectors of the channel covariance matrix for the specified block length and the gains of the individual channels are the eigenvalues an optimal bit allocation and energy distribution are derived for the set of parallel channels under an accurate extension of the continuous approximation for power in optimal multidimensional signal sets for constellations with unequal signal spacing in different dimensions examples are presented that demonstrate performance advantages with respect to zero forcing decision feedback methods that use the same coset code on the same partial response channel only resampling the channel at an optimal rate and assuming no errors in the feedback path will bring the performance of the decision feedback methods up to the level of the vector coded system

a priori bounds for co dimension one isometric embeddings let x inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i g inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i be a c 4 isometric embedding of a c4 metric g of nonnegative sectional curvature on inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 03i into the euclidean space inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 04i we prove a priori bounds for the trace of the second fundamental form h in terms of the scalar curvature r of g and the diameter d of the space inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 05i g these estimates give a bound on the extrinsic geometry in terms of intrinsic quantities they generalize estimates originally obtained by weyl for the case n 2 and positive curvature and then by p guan and the first author for nonnegative curvature and n 2 using c 2 interior estimates of evans and krylov for concave fully nonlinear elliptic partial differential equations these bounds allow us to obtain the following convergence theorem for any 0 the set of metrics of nonnegative sectional curvature and scalar curvature bounded below by which are isometrically embedable in euclidean space inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 06i is closed in the holder space c 4 0 suppose that g is a smooth metric of nonnegative sectional curvature and positive scalar curvature on inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 07i which admits locally convex isometric embedding into inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 08i does inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 09i g then admit a smooth global isometric embedding x inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 10i g inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 11i

weighted sum rate optimization for downlink multiuser mimo coordinated base station systems centralized and distributed algorithms this paper considers the joint linear transceiver design problem for the downlink multiuser multiple input multiple output mimo systems with coordinated base stations bss we consider maximization of the weighted sum rate with per bs antenna power constraint problem we propose novel centralized and computationally efficient distributed iterative algorithms that achieve local optimum to the latter problem these algorithms are described as follows first by introducing additional optimization variables we reformulate the original problem into a new problem second for the given precoder matrices of all users the optimal receivers are computed using minimum mean square error mmse method and the optimal introduced variables are obtained in closed form expressions third by keeping the introduced variables and receivers constant the precoder matrices of all users are optimized by using second order cone programming socp and matrix fractional minimization approaches for the centralized and distributed algorithms respectively finally the second and third steps are repeated until these algorithms converge we have shown that the proposed algorithms are guaranteed to converge we also show that the proposed algorithms require less computational cost than that of the existing linear algorithm all simulation results demonstrate that our distributed algorithm achieves the same performance as that of the centralized algorithm moreover the proposed algorithms outperform the existing linear algorithm in particular when each of the users has single antenna we have observed that the proposed algorithms achieve the global optimum

from elementary calculations to hall polynomials abstract via the computation of cokernels and using the theorem of krull remak and schmidt to decompose a given representation the task of computing hall polynomials for dynkin quivers can be reduced to solving many systems of linear equations the resulting method has been applied to obtain hall polynomials for a quiver of type e 8

hall polynomials for the representation finite hereditary algebras let k be a field let r be a finite dimensional k algebra with centre k which is representation finite and hereditary thus r is morita equivalent to the tensor algebra of a k species with underlying graph a a disjoint union of dynkin diagrams and the set of isomorphism classes of indecomposable r modules corresponds bijectively to the set of positive roots of the corresponding semisimple complex lie algebra g see g and drl consequently the grothendieck group k r mod of all finitely generated r modules modulo split exact sequences is the free abelian group with basis indexed by let h be a cartan subalgebra of g and g n h 0 n the corresponding triangular decomposition note that n is the direct sum of one dimensional complex vectorspaces indexed by the elements of so we may identify k r mod q and n as vectorspaces and we deal with the problem of how to recover the lie multiplication of n on k r mod we have shown in r2 that the grothendieck group k r mod may be considered in a natural way as a lie algebra by using as structure constants the evaluations of hall polynomials at 1 the aim of this paper is to show that this lie algebra k r mod can be identified with a chevalley h form of n in particular k r mod and n are isomorphic as lie algebras we are going to determine all possible polynomials which occur as hall polynomials p where x y z e there are precisely 16 different polynomials including the zero polynomial cpo and the absolute value of their evaluations at 1 is bounded by 3 one easily observes that q x 0 in case y z x thus let us assume y z x in this case precisely one of the two polynomials cp and cpzz is non zero the non zero polynomials cp can be written in the form irqi where c i t with 1 d r 6 3 and pi is one of the following 12 integral polynomials

composition algebras of affine type abstract the aim of this paper is to study the structure of the composition algebras of affine type it turns out that they have a triangular decomposition p t i corresponding to the division of the indecomposables into the preprojectives the regulars and the preinjectives by the recent ringel green theorem the composition algebra can be twisted in order to obtain the positive partu of the drinfeld jimbo quantized enveloping algebrau u u0 u of the corresponding kac moody algebra and one obtains a corresponding triangular decomposition foru in particular a natural basis ofu in terms ofa representations

applications of convex optimization in signal processing and digital communication in the last two decades the mathematical programming community has witnessed some spectacular advances in interior point methods and robust optimization these advances have recently started to significantly impact various fields of applied sciences and engineering where computational efficiency is essential this paper focuses on two such fields digital signal processing and communication in the past the widely used optimization methods in both fields had been the gradient descent or least squares methods both of which are known to suffer from the usual headaches of stepsize selection algorithm initialization and local minima with the recent advances in conic and robust optimization the opportunity is ripe to use the newly developed interior point optimization techniques and highly efficient software tools to help advance the fields of signal processing and digital communication this paper surveys recent successes of applying interior point and robust optimization to solve some core problems in these two fields the successful applications considered in this paper include adaptive filtering robust beamforming design and analysis of multi user communication system channel equalization decoding and detection throughout our emphasis is on how to exploit the hidden convexity convex reformulation of semi infinite constraints analysis of convergence complexity and performance as well as efficient practical implementation

network mimo with linear zero forcing beamforming large system analysis impact of channel estimation and reduced complexity scheduling we consider the downlink of a multicell system with multiantenna base stations and single antenna user terminals arbitrary base station cooperation clusters distance dependent propagation pathloss and general fairness requirements base stations in the same cooperation cluster employ joint transmission with linear zero forcing beamforming subject to sum or per base station power constraints intercluster interference is treated as noise at the user terminals analytic expressions for the system spectral efficiency are found in the large system limit where both the numbers of users and antennas per base station tend to infinity with a given ratio in particular for the per base station power constraint we find new results in random matrix theory yielding the squared frobenius norm of submatrices of the moore penrose pseudo inverse for the structured non i i d channel matrix resulting from the cooperation cluster user distribution and path loss coefficients the analysis is extended to the case of nonideal channel state information at the transmitters obtained through explicit downlink channel training and uplink feedback specifically our results illuminate the trade off between the benefit of a larger number of cooperating antennas and the cost of estimating higher dimensional channel vectors furthermore our analysis leads to a new simplified downlink scheduling scheme that preselects the users according to probabilities obtained from the large system results depending on the desired fairness criterion the proposed scheme performs close to the optimal finite dimensional opportunistic user selection while requiring significantly less channel state feedback since only a small fraction of preselected users must feed back their channel state information

hall algebras associated to triangulated categories by counting triangles and applying the octahedral axiom we find a direct way to prove the formula of toen in t section 5 1 for a triangulated category satisfying the left homological finiteness condition

domestic canonical algebras and simple lie algebras for each simply laced dynkin graph we realize the simple complex lie algebra of type as a quotient algebra of the complex degenerate composition lie algebra l a 1 mathbb c of a domestic canonical algebra a of type by some ideal i of l a 1 mathbb c that is defined via the hall algebra of a and give an explicit form of i moreover we show that each root space of l a 1 mathbb c i has a basis given by the coset of an indecomposable a module m with root easily computed by the dimension vector of m