comparison of distributed beamforming algorithms for mimo interference networks this paper presents a comparative study of algorithms for jointly optimizing beamformers and receive filters in an interference network where each node may have multiple antennas each user transmits at most one data stream and interference is treated as noise we focus on techniques that seek good suboptimal solutions by means of iterative and distributed updates those include forward backward iterative algorithms max signal to interference plus noise ratio sinr and interference leakage weighted sum mean squared error mse algorithms and interference pricing with incremental signal to noise ratio snr adjustments we compare their properties in terms of convergence and information exchange requirements and then numerically evaluate their sum rate performance averaged over random stationary channel realizations the numerical results show that the max sinr algorithm achieves the maximum degrees of freedom i e supports the maximum number of users with near zero interference and exhibits better convergence behavior at high snrs than the weighted sum mse algorithms however it assumes fixed power per user and achieves only a single point in the rate region whereas the weighted sum mse criterion gives different points in contrast the incremental snr algorithm adjusts the beam powers and deactivates users when interference alignment is infeasible furthermore that algorithm can provide a slight increase in sum rate relative to max sinr at the cost of additional iterations

complex valued matrix differentiation techniques and key results a systematic theory is introduced for finding the derivatives of complex valued matrix functions with respect to a complex valued matrix variable and the complex conjugate of this variable in the framework introduced the differential of the complex valued matrix function is used to identify the derivatives of this function matrix differentiation results are derived and summarized in tables which can be exploited in a wide range of signal processing related situations

regularity on abelian varieties i we introduce the notion of mukai regularity m regularity for coherent sheaves on abelian varieties the definition is based on the fourier mukai transform and in a special case depending on the choice of a polarization it parallels and strengthens the usual castelnuovo mumford regularity mukai regularity has a large number of applications ranging from basic properties of linear series on abelian varieties and defining equations for their subvarieties to higher dimensional type statements and to a study of special classes of vector bundles some of these applications are explained here while others are the subject of upcoming sequels

asynchronous iterative water filling for gaussian frequency selective interference channels this paper considers the maximization of information rates for the gaussian frequency selective interference channel subject to power and spectral mask constraints on each link to derive decentralized solutions that do not require any cooperation among the users the optimization problem is formulated as a static noncooperative game of complete information to achieve the so called nash equilibria of the game we propose a new distributed algorithm called asynchronous iterative water filling algorithm in this algorithm the users update their power spectral density psd in a completely distributed and asynchronous way some users may update their power allocation more frequently than others and they may even use outdated measurements of the received interference the proposed algorithm represents a unified framework that encompasses and generalizes all known iterative water filling algorithms e g sequential and simultaneous versions the main result of the paper consists of a unified set of conditions that guarantee the global converge of the proposed algorithm to the unique nash equilibrium of the game

mimo cognitive radio a game theoretical approach the concept of cognitive radio cr has recently received great attention from the research community as a promising paradigm to achieve efficient use of the frequency resource by allowing the coexistence of licensed primary and unlicensed secondary users in the same bandwidth in this paper we propose and analyze a totally decentralized approach based on game theory to design cognitive mimo transceivers who compete with each other to maximize their information rate the formulation incorporates constraints on the transmit power as well as null and or soft shaping constraints on the transmit covariance matrix so that the interference generated by secondary users be confined within the temperature interference limit required by the primary users we provide a unified set of conditions that guarantee the uniqueness and global asymptotic stability of the nash equilibrium of all the proposed games through totally distributed and asynchronous algorithms interestingly the proposed algorithms overcome the main drawback of classical waterfilling based algorithms the violation of the temperature interference limit and they have the desired features required for cr applications such as low complexity distributed implementation robustness against missing or outdated updates of the users and fast convergence behavior

on a risk model with stochastic premiums income and dependence between income and loss labbe and sendova 2009 9 consider a compound poisson risk model with stochastic premiums income in this paper we extend their model by assuming that there exists a specific dependence structure among the claim sizes interclaim times and premium sizes assume that the distributions of the premium sizes and interclaim times are controlled by the claim sizes when the individual premium sizes are exponentially distributed the laplace transforms and defective renewal equations for the gerber shiu discounted penalty functions are obtained when the individual premium sizes have rational laplace transforms we show that the laplace transforms for the discounted penalty functions can also be obtained

the mimo iterative waterfilling algorithm this paper considers the noncooperative maximization of mutual information in the vector gaussian interference channel in a fully distributed fashion via game theory this problem has been widely studied in a number of works during the past decade for frequency selective channels and recently for the more general multiple input multiple output mimo case for which the state of the art results are valid only for nonsingular square channel matrices surprisingly these results do not hold true when the channel matrices are rectangular and or rank deficient matrices the goal of this paper is to provide a complete characterization of the mimo game for arbitrary channel matrices in terms of conditions guaranteeing both the uniqueness of the nash equilibrium and the convergence of asynchronous distributed iterative waterfilling algorithms our analysis hinges on new technical intermediate results such as a new expression for the mimo waterfilling projection valid also for singular matrices a mean value theorem for complex matrix valued functions and a general contraction theorem for the multiuser mimo watefilling mapping valid for arbitrary channel matrices the quite surprising result is that uniqueness convergence conditions in the case of tall possibly singular channel matrices are more restrictive than those required in the case of full rank fat channel matrices we also propose a modified game and algorithm with milder conditions for the uniqueness of the equilibrium and convergence and virtually the same performance in terms of nash equilibria of the original game

an iteratively weighted mmse approach to distributed sum utility maximization for a mimo interfering broadcast channel consider the multiple input multiple output mimo interfering broadcast channel whereby multiple base stations in a cellular network simultaneously transmit signals to a group of users in their own cells while causing interference to each other the basic problem is to design linear beamformers that can maximize the system throughput in this paper we propose a linear transceiver design algorithm for weighted sum rate maximization that is based on iterative minimization of weighted mean square error mse the proposed algorithm only needs local channel knowledge and converges to a stationary point of the weighted sum rate maximization problem furthermore the algorithm and its convergence can be extended to a general class of sum utility maximization problem the effectiveness of the proposed algorithm is validated by numerical experiments

combining beamforming and orthogonal space time block coding multiple transmit and receive antennas can be used in wireless systems to achieve high data rate communication efficient space time codes have been developed that utilize a large portion of the available capacity these codes are designed under the assumption that the transmitter has no knowledge about the channel in this work on the other hand we consider the case when the transmitter has partial but not perfect knowledge about the channel and how to improve a predetermined code so that this fact is taken into account a performance criterion is derived for a frequency nonselective fading channel and then utilized to optimize a linear transformation of the predetermined code the resulting optimization problem turns out to be convex and can thus be efficiently solved using standard methods in addition a particularly efficient solution method is developed for the special case of independently fading channel coefficients the proposed transmission scheme combines the benefits of conventional beamforming with those given by orthogonal space time block coding simulation results for a narrow band system with multiple transmit antennas and one or more receive antennas demonstrate significant gains over conventional methods in a scenario with nonperfect channel knowledge

the expected discounted penalty function under a renewal risk model with stochastic income abstract in this paper we consider a renewal risk model with stochastic premiums income we assume that the premium number process and the claim number process are a poisson process and a generalized erlang n processes respectively when the individual stochastic premium sizes are exponentially distributed the laplace transform and a defective renewal equation for the gerber shiu discounted penalty function are obtained furthermore the discounted joint distribution of the surplus just before ruin and the deficit at ruin is given when the claim size distributions belong to the rational family the explicit expression of the gerber shiu discounted penalty function is derived finally a specific example is provided

the expected discounted penalty at ruin in the risk process with random income we extend the classical risk model to the case where the premium income process based on a poisson process is no longer a linear function we examine the expected discounted value of a penalty at ruin which is considered as a function of the initial surplus we derive a defective renewal equation satisfied by the discounted penalty function the solution of this renewal equation is then given the associated compound geometric distribution is also studied

the expected discounted penalty function under a risk model with stochastic income quantities of interest in ruin theory are investigated under the general framework of the expected discounted penalty function assuming a risk model where both premiums and claims follow compound poisson processes both a defective renewal equation and an integral equation satisfied by the expected discounted penalty function are established some implications that these equations have on particular quantities such as the discounted deficit and the probability of ultimate ruin are illustrated finally the case when premiums have erlang n b distribution and the distribution of the claims is arbitrary is investigated in more depth throughout the paper specific examples where claims and premiums have particular distributions are provided

gradient of mutual information in linear vector gaussian channels this paper considers a general linear vector gaussian channel with arbitrary signaling and pursues two closely related goals i closed form expressions for the gradient of the mutual information with respect to arbitrary parameters of the system and ii fundamental connections between information theory and estimation theory generalizing the fundamental relationship recently unveiled by guo shamai and verdu spl acute we show that the gradient of the mutual information with respect to the channel matrix is equal to the product of the channel matrix and the error covariance matrix of the best estimate of the input given the output gradients and derivatives with respect to other parameters are then found via the differentiation chain rule

optimized signaling for mimo interference systems with feedback the system mutual information of a multiple input multiple output mimo system with multiple users which mutually interfere is considered perfect channel state information is assumed to be known to both transmitters and receivers asymptotic performance analysis shows that the system mutual information changes behavior as the interference becomes sufficiently strong in particular beamforming is the optimum signaling for all users when the interference is large we propose several numerical approaches to decide the covariance matrices of the transmitted signals and compare their performance in terms of the system mutual information we model the system as a noncooperative game and perform iterative water filling to find the nash equilibrium distributively a centralized global approach and a distributed iterative approach based on the gradient projection method are also proposed numerical results show that all proposed approaches give better performance than the standard signaling which is optimum for the case without interference both the global and the iterative gradient projection methods are shown to outperform the nash equilibrium significantly

coexisting linear and widely linear transceivers in the mimo interference channel recent results have shown the benefits of widely linear precoding wlp in the mimo interference channel mimo ic assuming that all transmitters can follow the same strategy motivated by a transitional scenario where legacy linear transmitters coexist with widely linear ones this work investigates the general k user mimo ic in a heterogeneous linear and widely linear transmitter deployment in particular we address the maximization of the weighted sum rate wsr for widely linear transmit filters design through the use of the complex valued formulation since the maximum wsr problem is non convex and thus difficult to be solved we formulate an equivalent minimum weighted mean square error problem that allows deriving closed form expressions for widely linear transceivers then an iterative procedure is proposed which is proven to reach a stationary point of the maximum wsr problem simulations show that the proposed procedure allows increasing the sum rate as compared to coordinated linear transceiver schemes the gains are larger and significant in two different nonexclusive conditions as the interference level increases or when the number of antennas is low

robust mimo cognitive radio via game theory cognitive radio cr systems improve the spectral efficiency by allowing the coexistence in harmony of primary users pus the legacy users with secondary users sus this coexistence is built on the premises that no su can generate interference higher than some prescribed limits against pus the system design based on perfect channel state information csi can easily end up violating the interference limits in a realistic situation where csi may be imperfect in this paper we propose a robust design of cr systems composed of multiple pus and multiple noncooperative sus in either single input single output siso frequency selective channels or more general multiple input multiple output mimo channels we formulate the design of the su network as a noncooperative game where the sus compete with each other over the resources made available by the pus by maximizing their own information rates subject to the transmit power and robust interference constraints following the philosophy of the worst case robustness we take explicitly into account the imperfectness of su to pu csi by adopting proper interference constraints that are robust with respect to the worst channel errors relying on the variational inequality theory we study the existence and uniqueness properties of the nash equilibria of the resulting robust games and devise totally asynchronous and distributed algorithms along with their convergency properties we also propose efficient numerical methods based on decomposition techniques to compute the robust transmit strategy for each su

the constrained total least squares technique and its applications to harmonic superresolution the constrained total least squares ctls method is a natural extension of tls to the case when the noise components of the coefficients are algebraically related the ctls technique is developed and some of its applications to superresolution harmonic analysis are presented the ctls problem is reduced to an unconstrained minimization problem over a small set of variables a perturbation analysis of the ctls solution is derived and from it the root mean square error rmse of the ctls solution which is valid for small noise levels is obtained in closed form the complex version of the newton method is derived and applied to determine the ctls solution it is also shown that the ctls problem is equivalent to a constrained parameter maximum likelihood problem the ctls technique is applied to estimate the frequencies of sinusoids in white noise and the angle of arrival of narrowband wavefronts at a linear uniform array in both cases the ctls method shows superior or similar accuracy to other advanced techniques

pluricanonical maps of varieties of maximal albanese dimension let x be a smooth complex projective algebraic variety of maximal albanese dimension we give a characterization of kappa x in terms of the set v 0 x omega x p in mbox rm pic 0 x h 0 x omega x otimes p ne 0 an immediate consequence of this is that the kodaira dimension kappa x is invariant under smooth deformations we then study the pluricanonical maps varphi m x dashrightarrow mathbb p h 0 x mk x we prove that if x is of general type varphi m is generically finite for m geq 5 and birational for m geq 5 mbox rm dim x 1 more generally we show that for m geq 6 the image of varphi m is of dimension equal to kappa x and for m geq 6 kappa x 2 varphi m is the stable canonical map

irregular varieties with geometric genus one theta divisors and fake tori abstract we study the albanese image of a compact kahler manifold whose geometric genus is one in particular we prove that if the albanese map is not surjective then the manifold maps surjectively onto an ample divisor in some abelian variety and in many cases the ample divisor is a theta divisor with a further natural assumption on the topology of the manifold we prove that the manifold is an algebraic fiber space over a genus two curve finally we apply these results to study the geometry of a compact kahler manifold which has the same hodge numbers as those of an abelian variety of the same dimension

energy efficient spectrum sharing strategy selection for cognitive mimo interference channels in this paper we propose a promising discrete game theoretic framework for distributed energy efficient discrete spectrum sharing strategy selection i e joint discrete power control and multimode precoding strategy selection with limited feedback for cognitive mimo interference channels given the competitive nature the secondary users are assumed to be selfish and noncooperative each of whom attempts to maximize its individual energy efficiency under a minimum data rate constraint and an interference power constraint a mechanism for shutting down links is proposed to reduce interference and save energy a payoff function is designed to guarantee the feasibility of the pure strategy nash equilibrium with no need to know the infeasible strategy profiles a spectrum sharing strategy profile is said to be feasible if the stated constraints are satisfied otherwise the spectrum sharing strategy profile is said to be infeasible i e they may not satisfy the interference power constraint and minimum data rate constraint in advance we then investigate the existence and the feasibility of the pure strategy nash equilibrium and further devise a pricing based distributed algorithm for spectrum strategy selection the proposed algorithm is proved to converge to a feasible pure strategy nash equilibrium under specific conditions moreover by studying the relationship between our proposed game and the social optimum we find that the pricing mechanism can result in pareto improvement and lead to better convergence for the proposed distributed algorithm numerical results show that our designed algorithm significantly outperforms the random selection algorithm and the pricing mechanism has a dramatic effect in improving the system performance

deformation theory generic vanishing theorems and some conjectures of enriques catanese and beauville introduction 389 w notation and conventions 392 w deformations of cohomology groups 392 w generic vanishing criteria for topologically trivial line bundles 397 w a nakano type generic vanishing theorem 401 w an application to surfaces 405 references 407

gv sheaves fourier mukai transform and generic vanishing we prove a formal criterion for generic vanishing in the sense originated by green and lazarsfeld and pursued further by hacon but in the context of an arbitrary fourier mukai correspondence for smooth projective varieties we apply this to deduce a kodaira type generic vanishing theorem for adjoint bundles associated to nef line bundles and in fact a more general generic nadel type vanishing theorem for multiplier ideal sheaves still in the context of the picard variety the same method gives various other generic vanishing results by reduction to standard vanishing theorems we further use our criterion in order to address some examples related to generic vanishing on higher rank moduli spaces

optimal linear precoding strategies for wideband noncooperative systems based on game theory part i nash equilibria in this two part paper we propose a decentralized strategy based on a game theoretic formulation to find out the optimal precoding multiplexing matrices for a multipoint to multipoint communication system composed of a set of wideband links sharing the same physical resources i e time and bandwidth we assume as optimality criterion the achievement of a nash equilibrium and consider two alternative optimization problems 1 the competitive maximization of mutual information on each link given constraints on the transmit power and on the spectral mask imposed by the radio spectrum regulatory bodies and 2 the competitive maximization of the transmission rate using finite order constellations under the same constraints as above plus a constraint on the average error probability in this first part of the paper we start by showing that the solution set of both noncooperative games is always nonempty and contains only pure strategies then we prove that the optimal precoding multiplexing scheme for both games leads to a channel diagonalizing structure so that both matrix valued problems can be recast in a simpler unified vector power control game with no performance penalty thus we study this simpler game and derive sufficient conditions ensuring the uniqueness of the nash equilibrium interestingly although derived under stronger constraints incorporating for example spectral mask constraints our uniqueness conditions have broader validity than previously known conditions finally we assess the goodness of the proposed decentralized strategy by comparing its performance with the performance of a pareto optimal centralized scheme to reach the nash equilibria of the game in part ii we propose alternative distributed algorithms along with their convergence conditions

optimal linear precoding strategies for wideband non cooperative systems based on game theory part ii algorithms in this two part paper we address the problem of finding the optimal precoding multiplexing scheme for a set of noncooperative links sharing the same physical resources e g time and bandwidth we consider two alternative optimization problems p l the maximization of mutual information on each link given constraints on the transmit power and spectral mask and p 2 the maximization of the transmission rate on each link using finite order constellations under the same constraints as in p l plus a constraint on the maximum average error probability on each link aiming at finding decentralized strategies we adopted as optimality criterion the achievement of a nash equilibrium and thus we formulated both problems p l and p 2 as strategic noncooperative matrix valued games in part i of this two part paper after deriving the optimal structure of the linear transceivers for both games we provided a unified set of sufficient conditions that guarantee the uniqueness of the nash equilibrium in this part ii of the paper we focus on the achievement of the equilibrium and propose alternative distributed iterative algorithms that solve both games specifically the new proposed algorithms are the following 1 the sequential and simultaneous iterative waterfilling based algorithms incorporating spectral mask constraints and 2 the sequential and simultaneous gradient projection based algorithms establishing an interesting link with variational inequality problems our main contribution is to provide sufficient conditions for the global convergence of all the proposed algorithms which although derived under stronger constraints incorporating for example spectral mask constraints have a broader validity than the convergence conditions known in the current literature for the sequential iterative waterfilling algorithm

the gerber shiu discounted penalty function in the delayed renewal risk process with random income abstract this paper considers the delayed renewal risk model with random premium income the gerber shiu discounted penalty function in the delayed renewal model is expressed in terms of the corresponding gerber shiu function in the ordinary renewal model the obtained results can be viewed as the discrete analogy of the classical sparre anderson risk model

nash equilibrium seeking in noncooperative games we introduce a non model based approach for locally stable convergence to nash equilibria in static noncooperative games with n players in classical game theory algorithms each player employs the knowledge of the functional form of his payoff and the knowledge of the other players actions whereas in the proposed algorithm the players need to measure only their own payoff values this strategy is based on the extremum seeking approach which has previously been developed for standard optimization problems and employs sinusoidal perturbations to estimate the gradient we consider static games with quadratic payoff functions before generalizing our results to games with non quadratic payoff functions that are the output of a dynamic system specifically we consider general nonlinear differential equations with n inputs and n outputs where in the steady state the output signals represent the payoff functions of a noncooperative game played by the steady state values of the input signals we employ the standard local averaging theory and obtain local convergence results for both quadratic payoffs where the actual convergence is semi global and non quadratic payoffs where the potential existence of multiple nash equilibria precludes semi global convergence our convergence conditions coincide with conditions that arise in model based nash equilibrium seeking however in our framework the user is not meant to check these conditions because the payoff functions are presumed to be unknown for non quadratic payoffs convergence to a nash equilibrium is not perfect but is biased in proportion to the perturbation amplitudes and the higher derivatives of the payoff functions we quantify the size of these residual biases and confirm their existence numerically in an example noncooperative game in this example we present the first application of extremum seeking with projection to ensure that the players actions remain in a given closed and bounded action set

real and complex monotone communication games noncooperative game theoretic tools have been increasingly used to study many important resource allocation problems in communications networking smart grids and portfolio optimization in this paper we consider a general class of convex nash equilibrium problems neps where each player aims at solving an arbitrary smooth convex optimization problem differently from most of current works we do not assume any specific structure for the players problems and we allow the optimization variables of the players to be matrices in the complex domain our main contribution is the design of a novel class of distributed asynchronous best response algorithms suitable for solving the proposed neps even in the presence of multiple solutions the new methods whose convergence analysis is based on variational inequality vi techniques can select among all the equilibria of a game those that optimize a given performance criterion at the cost of limited signaling among the players this is a major departure from existing best response algorithms whose convergence conditions imply the uniqueness of the ne some of our results hinge on the use of vi problems directly in the complex domain the study of these new kind of vis also represents a noteworthy innovative contribution we then apply the developed methods to solve some new generalizations of single input single output siso and multiple input multiple output mimo games in cognitive radio systems showing a considerable performance improvement over classical pure noncooperative schemes

on the gerber shiu discounted penalty function in a risk model with two types of delayed claims and random income abstract this paper considers a risk model with random premium income and two types of by claims which is an extension to the general delayed claims process where there is only one type of by claim and the premium income is a constant assume that each main claim induces one of the two by claims the by claim and its associated main claim occur at the same time when the main claim amount is less than a threshold variable otherwise the occurrence of the by claim will be delayed an integral equations system for the gerber shiu discounted penalty functions is presented by using auxiliary risk models given that the premium size is exponentially distributed the explicit expression for the laplace transform of the gerber shiu penalty function is derived by applying rouche theorem the gerber shiu discounted penalty functions with reciprocal of the mean of premium surplus are obtained according to lagrange interpolation theorem we prove that the gerber shiu discounted penalty function satisfies a defective renewal equation finally under the assumption of the claim sizes satisfying exponential distribution the explicit formula of the ruin probability is derived when the discounted factor equals zero and the penalty function equals one

learning in an uncertain world mimo covariance matrix optimization with imperfect feedback in this paper we present a distributed learning algorithm for the optimization of signal covariance matrices in gaussian multiple input and multiple output mimo multiple access channel with imperfect and possibly delayed feedback the algorithm is based on the method of matrix exponential learning mxl and it has the same information and computation requirements as distributed water filling however unlike water filling the proposed algorithm converges to the system s optimum signal covariance profile even under stochastic uncertainty and imperfect feedback moreover the algorithm also retains its convergence properties in the presence of user update asynchronicities random delays and or ergodically changing channel conditions our theoretical analysis is complemented by extensive numerical simulations which illustrate the robustness and scalability of mxl in realistic network conditions in particular the algorithm retains its convergence speed even for large numbers of users and or antennas per user

a coordinate descent algorithm for complex joint diagonalization under hermitian and transpose congruences abstract this paper deals with the problem of joint complexmatrix diagonalization by considering both the hermitian andtranspose congruences we address the general case where thesearched diagonalizing matrix is apriori nonunitary based onthe minimization of a quadratic criterion we propose a ex ible algorithm in the sense that it allows to directly consider arectangular diagonalizing matrix and to take into considerationboth the hermitian and transpose congruences within the sameframework the proposed algorithm is a coordinate descentalgorithm that is based on an approximate criterion leading to theanalytical derivation of the minima arguments computer simu lations are drawn to illustrate the usefulness and performancesof the algorithm and a comparison to state of the art algorithmsis presented finally an application to independent componentanalysis based on fourth order statistics is also presented index terms blind source separation complex symmetric ma trices hermitian matrices independent component analysis jointdiagonalization

a derived category approach to generic vanishing we prove a generic vanishing theorem for coherent sheaves on an abelian variety over an algebraically closed field k when k c this implies a conjecture of green and lazarsfeld

decomposition by partial linearization parallel optimization of multi agent systems we propose a novel decomposition framework for the distributed optimization of general nonconvex sum utility functions arising naturally in the system design of wireless multi user interfering systems our main contributions are i the development of the first class of inexact jacobi best response algorithms with provable convergence where all the users simultaneously and iteratively solve a suitably convexified version of the original sum utility optimization problem ii the derivation of a general dynamic pricing mechanism that provides a unified view of existing pricing schemes that are based instead on heuristics and iii a framework that can be easily particularized to well known applications giving rise to very efficient practical jacobi or gauss seidel algorithms that outperform existing ad hoc methods proposed for very specific problems interestingly our framework contains as special cases well known gradient algorithms for nonconvex sum utility problems and many block coordinate descent schemes for convex functions

the gerber shiu function and the generalized cramer lundberg model abstract in this paper we extend some results in cramer 7 by considering the expected discounted penalty function as a generalization of the infinite time ruin probability we consider his ruin theory model that allows the claim sizes to take positive as well as negative values depending on the sign of these amounts they are interpreted either as claims made by insureds or as income from deceased annuitants respectively we then demonstrate that when the events arrival process is a renewal process the gerber shiu function satisfies a defective renewal equation subsequently we consider some special cases such as when claims have exponential distribution or the arrival process is a compound poisson process and annuity related income has erlang n distribution we are then able to specify the parameter and the functions involved in the above mentioned defective renewal equation

a newton like algorithm for complex variables with applications in blind equalization although the newton algorithm has been extended to the complex domain in different forms none of them seems to be directly applicable to blind equalization therefore the objective of this correspondence is to develop an algorithm for blind equalization in the complex domain we propose a newton like algorithm based on a complex taylor series stochastic newton like algorithms snla for two blind equalization cost functions are developed simulations show that the new algorithms perform slightly better than the self orthogonalizing algorithm