renaming in an asynchronous environment this paper is concerned with the solvability of the problem of processor renaming in unreliable completely asynchronous distributed systems fischer et al prove in 8 that nontrivial consensus cannot be attained in such systems even when only a single benign processor failure is possible in contrast this paper shows that problems of processor renaming can be solved even in the presence of up to t n 2 faulty processors contradicting the widely held belief that no nontrivial problem can be solved in such a system the problems deal with renaming processors so as to reduce the size of the initial name space when only uniqueness of the new names is required we present a lower bound of n 1 on the size of the new name space and a renaming algorithm that establishes an upper bound on n t if the new names are required also to preserve the original order a tight bound of 2 n t 1 1 is obtained

power law shot noise the behavior of power law shot noise for which the associated impulse response functions assume a decaying power law form is explored expressions are obtained for the moments moment generating functions amplitude probability density functions autocorrelation functions and power spectral densities for a variety of parameters of the process for certain parameters the power spectral density exhibits 1 f type behavior over a substantial range of frequencies so that the process serves as a source of 1 f sup alpha shot noise for alpha in the range 0

downlink cellular network analysis with multi slope path loss models existing cellular network analyses and even simulations typically use the standard path loss model where received power decays like vert x vert alpha over a distance vert x vert this standard path loss model is quite idealized and in most scenarios the path loss exponent alpha is itself a function of vert x vert typically an increasing one enforcing a single path loss exponent can lead to orders of magnitude differences in average received and interference powers versus the true values in this paper we study multi slope path loss models where different distance ranges are subject to different path loss exponents we focus on the dual slope path loss function which is a piece wise power law and continuous and accurately approximates many practical scenarios we derive the distributions of sir snr and finally sinr before finding the potential throughput scaling which provides insight on the observed cell splitting rate gain the exact mathematical results show that the sir monotonically decreases with network density while the converse is true for snr and thus the network coverage probability in terms of sinr is maximized at some finite density with ultra densification network density goes to infinity there exists a phase transition in the near field path loss exponent alpha 0 if alpha 0 1 unbounded potential throughput can be achieved asymptotically if alpha 0 ultra densification leads in the extreme case to zero throughput

cell edge aware precoding for downlink massive mimo cellular networks we propose a cell edge aware cea zero forcing zf precoder that exploits the excess spatial degrees of freedom provided by a large number of base station bs antennas to suppress inter cell interference at the most vulnerable user equipments ues we evaluate the downlink performance of cea zf as well as that of a conventional cell edge unaware ceu zf precoder in a network with a random bs topology our analysis and simulations show that the proposed cea zf precoder outperforms ceu zf precoding in terms of i aggregate per cell data rate ii coverage probability and iii 95 likely or edge user rate in particular when both perfect channel state information and a large number of antennas n are available at the bss we demonstrate that the outage probability under cea zf and ceu zf decay as 1 n 2 and 1 n respectively this result identifies cea zf as a more effective precoding scheme for massive multiple input multiple output mimo cellular networks our framework also reveals the importance of scheduling the optimal number of ues per bs and confirms the necessity to control the amount of pilot contamination received during the channel estimation phase

small systems of generators of isotropic unitary groups over finite fields of characteristic not two abstract some sets of generators of unitary groups on a nonsingular hermitian space with hyperbolic rank at least one over a finite field of characteristic not 2 with an involution are studied all unitary groups are shown to be generated by 3 elements and all symplectic groups are generated by 2 elements

optimal selective feedback policies for opportunistic beamforming this paper studies the structure of downlink sum rate maximizing selective decentralized feedback policies for opportunistic beamforming under finite feedback constraints on the average number of mobile users feeding back first it is shown that any sum rate maximizing selective decentralized feedback policy must be a threshold feedback policy this result holds for all fading channel models with continuous distribution functions second the resulting optimum threshold selection problem is analyzed in detail this is a nonconvex optimization problem over finite dimensional euclidean spaces by utilizing the theory of majorization an underlying schur concave structure in the sum rate function is identified and the sufficient conditions for the optimality of homogenous threshold feedback policies are obtained applications of these results are illustrated for well known fading channel models such as rayleigh nakagami and rician fading channels rather surprisingly it is shown that using the same threshold value at all mobile users is not always a rate wise optimal feedback strategy even for a network in which mobile users experience statistically the same channel conditions for the rayleigh fading channel model on the other hand homogenous threshold feedback policies are proven to be rate wise optimal if multiple orthonormal data carrying beams are used to communicate with multiple mobile users simultaneously

optimal byzantine resilient convergence in uni dimensional robot networks given a set of robots with arbitrary initial location and no agreement on a global coordinate system convergence requires that all robots asymptotically approach the exact same but unknown beforehand location robots are oblivious they do not recall the past computations and are allowed to move in a one dimensional space additionally robots cannot communicate directly instead they obtain system related information only via visual sensors even though convergence and the classical distributed approximate agreement problem that requires correct processes to decide for some constant e values distance e apart and within the range of initial proposed values are similar we provide evidence that solving convergence in robot networks requires specific assumptions about synchrony and byzantine resilience in more detail we prove necessary and sufficient conditions for the convergence of mobile robots despite a subset of them being byzantine i e they can exhibit arbitrary behavior additionally we propose two deterministic convergence algorithms for robot networks and analyze their correctness and complexity in various atomicity and synchrony settings the first algorithm tolerates f byzantine robots for 2f 1 sized robot networks in fully synchronous atom networks while the second proposed algorithm tolerates f byzantine robots for 3f 1 sized robot networks in non atomic corda networks the resilience of these two algorithms is proved to be optimal

reduced feedback schemes using random beamforming in mimo broadcast channels a random beamforming scheme for the gaussian mimo broadcast channel with channel quality feedback is investigated and extended considering the case where the n receivers each have n receive antennas the effects of feeding back various amounts of signal to interference plus noise ratio sinr information are analyzed using the results from order statistics of the ratio of a linear combination of exponential random variables the distribution function of the maximum order statistic of the sinr observed at the receiver is found the analysis from viewing each antenna as an individual user is extended to allow combining at the receivers where it is known that the linear mmse combiner is the optimal linear receiver and the cdf for the sinr after optimal combining is derived analytically using the delta method the asymptotic distribution of the maximum order statistic of the sinr with and without combining is shown to be in the nomenclature of extreme order statistics of type 3 the throughput of the feedback schemes are shown to exhibit optimal scaling asymptotically in the number of users finally to further reduce the amount of feedback a hard threshold is applied to the sinr feedback the amount of feedback saved by implementing a hard threshold is determined and the effect on the system throughput is analyzed and bounded

gaussian approximation for the wireless multi access interference distribution the problem of gaussian approximation for the wireless multi access interference distribution in large spatial wireless networks is addressed first a principled methodology is presented to establish rates of convergence of the multi access interference distribution to a gaussian distribution for general bounded and power law decaying path loss functions the model is general enough to also include various random wireless channel dynamics such as fading and shadowing arising from multi path propagation and obstacles existing in the communication environment it is shown that the wireless multi access interference distribution converges to the gaussian distribution with the same mean and variance at a rate 1 where 0 is a parameter controlling the intensity of the planar possibly non stationary poisson point process generating node locations an explicit expression for the scaling coefficient is obtained as a function of fading statistics and the path loss function second an extensive numerical and simulation study is performed to illustrate the accuracy of the derived gaussian approximation bounds a good statistical fit between the interference distribution and its gaussian approximation is observed for moderate to high values of

analyzing the impact of access point density on the performance of finite area networks assuming a network of infinite extent several researchers have analyzed small cell networks using a poisson point process ppp location model leading to simple analytic expressions the general assumption has been that these results apply to finite area networks as well however do the results of infinite area networks apply to finite area networks in this paper we answer this question by obtaining an accurate approximation for the achievable signal to interference plus noise ratio sinr and user capacity in the downlink of a finite area network with a fixed number of access points aps the aps are uniformly distributed within the area of interest our analysis shows that crucially the results of infinite area networks are very different from those for finite area networks of low to medium ap density comprehensive simulations are used to illustrate the accuracy of our analysis for practical values of signal transmit powers and ap densities the analytic expressions capture the behavior of the system well as an added benefit the formulations developed here can be used in parametric studies for network design here the analysis is used to obtain the required number of aps to guarantee a desired target capacity in a finite area network

new combinatorial topology bounds for renaming the upper bound in the renaming task n 1 processes start with unique input names from a large space and must choose unique output names taken from a smaller name space 0 1 k to rule out trivial solutions a protocol must be anonymous the value chosen by a process can depend on its input name and on the execution but not on the specific process id attiya et al 1990 showed that renaming has a wait free solution when k 2n several algebraic topology proofs of a lower bound stating that no such protocol exists when k

multiplicative and transitive symplectic groups classification des sous groupes transitifs et multiplicatifs de sp v decomposition de sp v quand i ou r est stable

design modeling and performance analysis of multi antenna heterogeneous cellular networks this paper presents a stochastic geometry based framework for the design and analysis of downlink multi user multiple input multiple output mimo heterogeneous cellular networks with linear zero forcing transmit precoding and receive combining assuming rayleigh fading channels and perfect channel state information the generalized tiers of base stations may differ in terms of their poisson point process spatial density number of transmit antennas transmit power artificial biasing weight and number of user equipments served per resource block the spectral efficiency of a typical user equipped with multiple receive antennas is characterized using a non direct moment generating function based methodology with closed form expressions of the useful received signal and aggregate network interference statistics systematically derived in addition the area spectral efficiency is formulated under different space division multiple access and single user beamforming transmission schemes we examine the impact of different cellular network deployments propagation conditions antenna configurations and mimo setups on the achievable performance through theoretical and simulation studies based on the state of the art system parameters the results highlight the inherent limitations of baseline single input single output transmission and conventional sparse macro cell deployment as well as the promising potential of multi antenna communications and small cell solution in interference limited cellular environments

a non topological proof for the impossibility of k set agreement in the k set agreement task each process proposes a value and each correct process has to decide a value which was proposed so that at most k distinct values are decided using topological arguments it has been proved that k set agreement is unsolvable in the asynchronous wait free read write shared memory model when k

asymptotic analysis of reduced feedback strategies for mimo gaussian broadcast channels achieving the sum capacity of a multiple input multiple output mimo gaussian broadcast channel is known to require full channel state information csi at the base station which implies the need of a large amount of feedback information from the users different asymptotics of the sum capacity such as its scaling law with respect to the number of users n or the multiplexing gain are conventionally used to assess the performance of suboptimal schemes with reduced feedback or equivalently with partial channel state information at the transmitter in this correspondence the optimal scaling law of the sum rate with respect to n for fixed signal to noise ratio snr fixed number of transmit antennas m and any number of receiving antennas n i e m log log nn is proved to be achievable with a deterministic feedback of only one bit per user moreover the amount of feedback is shown to be further reduced with no asymptotic optimality loss by applying the selective feedback principle leading to an average feedback rate that scales as log n finally the asymptotic performance with respect to snr is studied by assessing how fast the number of users needs to increase with the snr in order to guarantee a noninterference limited behavior

on the user selection for mimo broadcast channels in this paper a downlink communication system in which a base station bs equipped with antennas communicates with users each equipped with receive antennas is considered an efficient suboptimum algorithm is proposed for selecting a set of users in order to maximize the sum rate throughput of the system in a rayleigh fading environment for the asymptotic case when tends to infinity the necessary and sufficient conditions in order to achieve the maximum sum rate throughput such that the difference between the achievable sum rate and the maximum value approaches zero is derived the complexity of our algorithm is investigated in terms of the required amount of feedback from the users to the bs as well as the number of searches required for selecting the users it is shown that the proposed method is capable of achieving a large portion of the sum rate capacity with a very low complexity

success probability and area spectral efficiency in multiuser mimo hetnets we derive a general and closed form result for the success probability in downlink multiple antenna mimo heterogeneous cellular networks hetnets utilizing a novel toeplitz matrix representation this main result which is equivalently the signal to interference ratio sir distribution includes multiuser mimo single user mimo and per tier biasing for k different tiers of randomly placed base stations bss assuming zero forcing precoding and perfect channel state information the large sir limit of this result admits a simple closed form that is accurate at moderate sirs e g above 5 db these results reveal that the sir invariance property of siso hetnets does not hold for mimo hetnets instead the success probability may decrease as the network density increases we prove that the maximum success probability is achieved by activating only one tier of bss while the maximum area spectral efficiency ase is achieved by activating all the bss this reveals a unique tradeoff between the ase and link reliability in multiuser mimo hetnets to achieve the maximum ase while guaranteeing a certain link reliability we develop efficient algorithms to find the optimal bs densities it is shown that as the link reliability requirement increases more bss and more tiers should be deactivated

a tractable framework for the analysis of dense heterogeneous cellular networks this paper investigates the downlink performance of dense k tier heterogeneous cellular networks hcns under general settings first gaussian approximation bounds for the standardized aggregate wireless interference awi in dense k tier hcns are obtained for when base stations bss in each tier are distributed over the plane according to a spatial and general poisson point process the kolmogorov smirnov ks distance is used to measure deviations of the distribution of the standardized awi from the standard normal distribution an explicit and analytical expression bounding the ks distance between these two distributions is obtained as a function of a broad range of network parameters such as per tier transmission power levels per tier bs intensity bs locations general fading statistics and general bounded path loss models bounds achieve a good statistical match between the standardized awi distribution and its normal approximation even for moderately dense hcns second various spatial performance metrics of interest such as outage capacity ergodic capacity and area spectral efficiency in the downlink of k tier hcns for general signal propagation models are investigated by making use of the derived distribution approximation results considering two specific bs association policies it is shown that the derived performance bounds track the actual performance metrics reasonably well for a wide range of bs intensities with the gap among them becoming negligibly small for denser hcn deployments finally both analytical and numerical results on the area spectral efficiency reveal a non linear growth trend with diminishing returns of hcn performance hence the sf sir invariance property does not hold under bounded path loss models which is a critical finding from the hcn design perspective in particular it points out a critical bs density beyond which the hcn performance starts to decline due to excessive wireless interference

modeling heterogeneous network interference using poisson point processes cellular systems are becoming more heterogeneous with the introduction of low power nodes including femtocells relays and distributed antennas unfortunately the resulting interference environment is also becoming more complicated making evaluation of different communication strategies challenging in both analysis and simulation leveraging recent applications of stochastic geometry to analyze cellular systems this paper proposes to analyze downlink performance in a fixed size cell which is inscribed within a weighted voronoi cell in a poisson field of interferers a nearest out of cell interferer out of cell interferers outside a guard region and cross tier interferers are included in the interference calculations bounding the interference power as a function of distance from the cell center the total interference is characterized through its laplace transform an equivalent marked process is proposed for the out of cell interference under additional assumptions to facilitate simplified calculations the interference distribution is approximated using the gamma distribution with second order moment matching the gamma approximation simplifies calculation of the success probability and average rate incorporates small scale and large scale fading and works with co tier and cross tier interference simulations show that the proposed model provides a flexible way to characterize outage probability and rate as a function of the distance to the cell edge

a comparison of time sharing dpc and beamforming for mimo broadcast channels with many users in this letter we derive the scaling laws of the sum rate for fading multiple input multiple output gaussian broadcast channels using time sharing to the strongest user dirty paper coding dpc and beamforming when the number of users receivers n is large throughout the letter we assume a fix average transmit power and consider a block fading rayleigh channel first we show that for a system with m transmit antennas and users equipped with n antennas the sum rate scales like mloglognn for dpc and beamforming when m is fixed and for any n either growing to infinity or not on the other hand when both m and n are fixed the sum rate of time sharing to the strongest user scales like min m n loglogn therefore the asymptotic gain of dpc over time sharing for the sum rate is m min m n when m and n are fixed it is also shown that if m grows as logn the sum rate of dpc and beamforming will grow linearly in m but with different constant multiplicative factors in this region the sum rate capacity of time sharing scales like nloglogn

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

reaching approximate agreement in the presence of faults this paper considers a variant of the byzantine generals problem in which processes start with arbitrary real values rather than boolean values or values from some bounded range and in which approximate rather than exact agreement is the desired goal algorithms are presented to reach approximate agreement in asynchronous as well as synchronous systems the asynchronous agreement algorithm is an interesting contrast to a result of fischer et al who show that exact agreement with guaranteed termination is not attainable in an asynchronous system with as few as one faulty process the algorithms work by successive approximation with a provable convergence rate that depends on the ratio between the number of faulty processes and the total number of processes lower bounds on the convergence rate for algorithms of this form are proved and the algorithms presented are shown to be optimal

more choices allow more faults set consensus problems in totally asynchronous systems abstract we define the k set consensus problem as an extension of the consensus problem where each processor decides on a single value such that the set of decided values in any run is of size at most k we require the agreement condition that all values decided upon are initial values of some processor we show that the problem has a simple k 1 resilient protocol in a totally asynchronous system in an attempt to come up with a matching lower bound on the number of failures we study the uncertainty condition which requires that there must be some initial configuration from which all possible input values can be decided we prove using a combinatorial argument that any k resilient protocol for the k set agreement problem would satisfy the uncertainty condition while this is not true for any k 1 resilient protocol this result seems to strengthen the conjecture that there is no k resilient protocol for this problem we prove this result for a restricted class of protocols our motivation for studying this problem is to test whether the number of choices allowed to the processors is related to the number of faults we hope that this will provide intuition towards achieving better bounds for more practical problems that arise in distributed computing e g the renaming problem the larger goal is to characterize the boundary between possibility and impossibility in asynchronous systems given multiple faults

reaching approximate byzantine consensus with multi hop communication abstract we are interested in approximate byzantine consensus problem wherein all the fault free processes reach consensus asymptotically approximately in finite time in particular we focus on the algorithms under which each process communicates with other processes that are up to l hops away and maintains minimal states across iterations for a given l we identify a necessary and sufficient condition on the network structure for the existence of iterative algorithms of interest our necessary and sufficient condition generalizes the tight condition identified in prior work for l 1 for l l where l is the length of a longest cycle free path in the given network our condition is equivalent to the necessary and sufficient conditions for exact consensus in undirected and directed networks both

generation of symplectic groups by transvections over local rings with at least 3 residue classes let r be a commutative local ring with identity 1 and a unique maximal ideal a let v denote a free module of rank iz over r with an alternating bilinear form f vx v r we assume r a f however v may have rad v qr a linear automorphism e of v is called an isometry on v if t satisfies f ax oy f x y for all x y in v the set of all isometries is a subgroup of the automorphism group of v which is called a symplectic group denoted by sp v for a submodule u of v we define u u e vi f v u 0 and rad u un u then u is called the orthogonal complement of u and rad u the radical of u for submodules u and w of v by u i w we mean u w with f u w 0 we define

the combinatorial structure of wait free solvable tasks this paper presents a self contained study of wait free solvable tasks a new necessary condition for wait free solvability based on a restricted set of executions is proved this set of executions induces a very simple to understand structure which is used to prove tight bounds for k set consensus and renaming the framework is based on topology but uses only elementary combinatorics and in contrast to previous works does not rely on algebraic or geometric arguments

exact sum rate analysis of mimo broadcast channels with random unitary beamforming random unitary beamforming rub is an attractive transmission scheme for mimo broadcast channels because of its ability to achieve high sum rate capacity with limited feedback in this letter we derive the exact analytical expressions for the ergodic sum rate of mimo broadcast channels with rub the analysis is facilitated by the development of the complete statistical characterization of ordered beam sinrs for a user which can find their application in many related problems

are wait free algorithms fast the time complexity of wait free algorithms in normal executions where no failures occur and processes operate at approximately the same speed is considered a lower bound of log n on the time complexity of any wait free algorithm that achieves approximate agreement among n processes is proved in contrast there exists a non wait free algorithm that solves this problem in constant time this implies an o log n time separation between the wait free and non wait free computation models on the positive side we present an o log n time wait free approximate agreement algorithm the complexity of this algorithm is within a small constant of the lower bound

spatial interference cancellation for multiantenna mobile ad hoc networks interference between nodes is a critical impairment in mobile ad hoc networks this paper studies the role of multiple antennas in mitigating such interference specifically a network is studied in which receivers apply zero forcing beamforming to cancel the strongest interferers assuming a network with poisson distributed transmitters and independent rayleigh fading channels the transmission capacity is derived which gives the maximum number of successful transmissions per unit area mathematical tools from stochastic geometry are applied to obtain the asymptotic transmission capacity scaling and characterize the impact of inaccurate channel state information csi it is shown that if each node cancels interferers the transmission capacity decreases as as the outage probability vanishes for fixed as grows the transmission capacity increases as where is the path loss exponent moreover csi inaccuracy is shown to have no effect on the transmission capacity scaling as vanishes provided that the csi training sequence has an appropriate length which we derive numerical results suggest that canceling merely one interferer by each node may increase the transmission capacity by an order of magnitude or more even when the csi is imperfect

on optimal downlink coverage in poisson cellular networks with power density constraints this paper studies downlink coverage maximization for cellular networks in which base station bs locations are modeled using a spatial poisson point process considering three different coverage models and under constraints on transmit power bs density and transmit power density firstly the coverage optimization problem is solved analytically for the first coverage model that focuses on noise limited communication by ignoring interference and random fading effects this model provides useful insights into the significance of bounded path loss models to obtain meaningful solutions for this problem the other two coverage models are based on the users received signal to interference plus noise ratio sinr from their associated bss for these models it is shown that the coverage optimization problem can be reduced to a constrained single dimensional optimization problem without any loss of optimality the related solutions can be obtained with limited computational complexity by resorting to a numerical search over a compact subset of candidate values bounds on the optimum bs density are also provided to further truncate the search space all results are derived for general bounded path loss models specific applications are also illustrated to provide further design insights and to highlight the importance of using bounded path loss models for coverage analysis

groups generated by symplectic transvections over local rings abstract let r be a commutative local ring with the unique maximal ideal a let v be a free module of rank n over r and let spn v be the symplectic group on v with an alternating bilinear form f v v r we study the generation of a subgroup tr m of spn v where m x v f x v r and tr m is defined as the subgroup generated by all symplectic transvections with axis x for x m our main goal is to get a nice necessary and sufficient condition for any subset n m satisfying tr n tr m where tr n is the group generated by all symplectic transvections with axis x for x n in particular if f is nonsingular we have tr m spn v and therefore our necessary and sufficient condition gives us a criterion for an arbitrarily given n m satisfying tr n spn v also we shall investigate the tr n orbit of each x m find some small sets of generators of tr m consisting of transvections in tr n and as a result solve the author s conjecture in generators and relations in groups and geometries a barlotti et al eds pp 47 67 proc nato asi c

iterative best beam selection for random unitary beamforming due to multi user diversity the performance of a random unitary beamforming rub can be improved upon in proportion to the number of users at the expense of an increase in feedback it is essential to identify a mechanism to determine a set of orthonormal beams for improving the channel capacity this paper proposes a unique scheme named iterative best beam selection based rub i rub in which the base station selects the best beamforming vectors from unitary matrices generated by the proposed iterative algorithm using the selected beamforming vector at each stage the beam set can ultimately be constructed for high sum rate capacity of downlink which leads to a reduction in feedback compared to those of other rub schemes based on the beam set selection

a layered analysis of consensus this paper introduces a simple notion of layering as a tool for analyzing well behaved runs of a given model of distributed computation using layering a model independent analysis of the consensus problem is performed and then applied to proving lower bounds and impossibility results for consensus in a number of familiar and less familiar models the proofs are simpler and more direct than existing ones and they expose a unified structure to the difficulty of reaching consensus in particular the proofs for the classical synchronous and asynchronous models now follow the same outline a new notion of connectivity among states in runs of a consensus protocol called potence connectivity is introduced this notion is more general than previous notions of connectivity used for this purpose and plays a key role in the uniform analysis of consensus

unitary groups with excellent s and entire e u l abstract unitary groups of hermitian forms with a hyperbolic rank at least one over local rings have been studied by d g james j algebra 52 1978 354 363 using his methods we extend some of his results i e generators congruence subgroups and the classification of subgroups normalized by the eichler subgroups to more general rings which contain semilocal rings

computation of shot noise probability distributions and densities the computation of the cumulative distribution cdf the complementary cdf ccdf and the density of certain shot noise random variables is discussed after subtracting off a few terms that can be computed in closed form what remains can be approximated by a general method for approximating samples of a cdf or ccdf by summing a fourier series whose coefficients are modulated samples of their characteristic function to approximate the density a spline is fit to the cdf samples and then differentiated when the density has corners it is important that the spline have coincident knots at these locations for shot noise densities these locations are easily identified