Search

This paper identifies the CEO characteristics that have an impact on the performance of family businesses listed in the Euronext in the post-COVID 19 period. CEO characteristics are evaluated on two dimensions, i.e., personal characteristics and corporate governance mechanisms. A sample of 137 firm-year observations from Portugal, Luxembourg, the Netherlands, Ireland, France, and Belgium was chosen. CEO attributes of age, gender, education, and family membership were combined with corporate governance mechanisms of ownership concentration, CEO duality, CEO directorships, and CEO tenure, to predict return on assets and return on equity, using OLS regression. GMM estimation and Two-Stage Least Squares were employed to establish the robustness of the results. Among CEO personal characteristics, CEO family membership has a positive impact on return on assets, and a positive impact on return on equity. Among corporate governance mechanisms, CEO duality had a negative impact on return on assets, and a negative impact on return on equity. CEO ownership, and CEO tenure had a positive impact on return on assets, and a positive impact on return on equity. This paper's value lies in its evaluation of the under-researched area of family businesses of Euronext-listed firms. It can be used by family businesses in the region, for the selection and training of CEOs to fulfill the goal of achieving superior financial performance. [ABSTRACT FROM AUTHOR]

This study aims to find the potential of Bus Rapid Transit (BRT) to attract the vehicle owners from their personal vehicles i.e., motorcars and motorcycles. Stated preference survey (questionnaires) and interviews were conducted at BRT (Metro Bus) Stations for the prediction of the individuals shifting from their private vehicles to BRT. Questions were designed critically as per the requirements of the research related to numerous aspects of BRT use i.e., vehicle ownership of the travelers, driving license holder, demographic characteristics, choice to use BRT if the fare increases, trip purpose and their prior mode of transportation for the same trip. A total of 374 responses, as per the population of the study area (Islamabad-Rawalpindi, Pakistan), were collected. The Multinomial Logistic Regression (MNL) model has been employed for four categories of vehicle ownerships i.e., “Car owners using BRT”, “Bike owners using BRT”, “Both Car and Bike owners using BRT” and the last one which has been taken as reference category is “BRT users with no vehicle ownership”. The analysis indicated that BRT has attracted considerably private vehicle users specially the bike owners. Some socio-economic factors like income and residence location (accessibility) additionally have a major effect on the selection of BRT. In addition, it has been observed that fare increase can alter the mode choice of the BRT users and they will again prefer their own vehicles. The Travel choice model developed in the study can be very useful for policy makers and transport planners to enhance the BRT service and attraction, to mitigate traffic congestion and car ownership.

Blockchain technology has emerged as a transformative solution for enhancing transparency and accountability across various dimensions of sustainable development. Given the limited exploration of Blockchain's synergy with sustainable development goals, there exists a compelling rationale to undertake a bibliometric literature review aimed at identifying forthcoming research trajectories. Employing the "Blockchain" AND "Sustainable Development Goals" keywords, an extensive search was conducted across Scopus and Web of Science databases. This endeavor yielded a corpus of 271 articles published within the span of 2017 to 2023. Subsequent adherence to inclusion and exclusion criteria refined this collection to 183 documents for meticulous Bibliographic Analysis through the Biblioshiny application of the Bibliometrix R package. The study charts a promising trajectory characterized by upward trends in recent years, unveiling the most prolific authors, productive nations, frequency of keywords, thematic evaluation, and collaborative endeavors. Delving into the systematic literature review, the research introduces five discernible future trends, along with a comprehensive research framework designed to amplify the generalizability of findings to pertinent stakeholders. [ABSTRACT FROM AUTHOR]

Predicting trends in the stock market is becoming complex and uncertain. In response, various artificial intelligence solutions have emerged. A significant solution for predicting the trends of a stock's volatile and chaotic nature is drawn from deep learning. The present study's objective is to compare and predict the closing price of the NIFTY 50 index through two significant deep learning methods—long short-term memory (LSTM) and backward elimination LSTM (BE-LSTM)—using 15 years' worth of per day data obtained from Bloomberg. This study has considered the variables of date, high, open, low, close volume, as well as the 14-period relative strength index (RSI), to predict the closing price. The results of the comparative study show that backward elimination LSTM performs better than the LSTM model for predicting the NIFTY 50 index price for the next 30 days, with an accuracy of 95%. In conclusion, the proposed model has significantly improved the prediction of the NIFTY 50 index price. [ABSTRACT FROM AUTHOR]

The distributed matrix multiplication problem with an unknown number of stragglers is considered, where the goal is to efficiently and flexibly obtain the product of two massive matrices by distributing the computation across $N$ servers. There are up to $N - R$ stragglers but the exact number is not known a priori. Motivated by reducing the computation load of each server, a flexible solution is proposed to fully utilize the computation capability of available servers. The computing task for each server is separated into several subtasks, constructed based on Entangled Polynomial codes by Yu et al. The final results can be obtained from either a larger number of servers with a smaller amount of computation completed per server or a smaller number of servers with a larger amount of computation completed per server. The required finite field size of the proposed solution is less than $2N$. Moreover, the optimal design parameters such as the partitioning of the input matrices are discussed. Our constructions can also be generalized to other settings such as batch distributed matrix multiplication and secure distributed matrix multiplication. [ABSTRACT FROM AUTHOR]

A linear computation broadcast (LCBC) setting is comprised of data that is stored at a server and $K$ users who wish to retrieve their respective desired linear functions of the data, while utilizing knowledge of certain other linear functions available to each user as prior side-information. The goal is to determine the smallest amount of broadcast information from the server that will simultaneously satisfy all the users’ demands. The reciprocal of the optimal broadcast cost is called the capacity of LCBC. The capacity is known for the 2-user setting, and recent studies of 3-user LCBC have revealed new insights into the need for sumset inequalities for LCBC. This letter introduces a generalization of the LCBC problem, where instead of storing all the data, the server may have partial knowledge in the form of some arbitrary linear function of the data. Building upon recent advances, the exact capacity is found for 2-user linear computation broadcast with partially informed server. [ABSTRACT FROM AUTHOR]

Motivated by recent interest in federated submodel learning, this work explores the fundamental problem of privately reading from and writing to a database comprised of $K$ files (submodels) that are stored across $N$ distributed servers according to an $X$ -secure threshold secret sharing scheme. One after another, various users wish to retrieve their desired file, locally process the information and then update the file in the distributed database while keeping the identity of their desired file private from any set of up to $T$ colluding servers. The availability of servers changes over time, so elastic dropout resilience is required. The main contribution of this work is an adaptive scheme, called ACSA-RW, that takes advantage of all currently available servers to reduce its communication costs, fully updates the database after each write operation even though the database is only partially accessible due to server dropouts, and ensures a memoryless operation of the network in the sense that the storage structure is preserved and future users may remain oblivious of the past history of server dropouts. The ACSA-RW construction builds upon cross-subspace alignment (CSA) codes that were originally introduced for $X$ -secure $T$ -private information retrieval and have been shown to be natural solutions for secure distributed batch matrix multiplication problems. ACSA-RW achieves the desired private read and write functionality with elastic dropout resilience, matches the best results for private-read from PIR literature, improves significantly upon available baselines for private-write, reveals a striking symmetry between upload and download costs, and exploits storage redundancy to accommodate arbitrary read and write dropout servers up to certain threshold values. It also answers in the affirmative an open question by Kairouz et al. for the case of partially colluding servers (i.e., tolerating collusion up to a threshold) by exploiting synergistic gains from the joint design of private read and write operations. [ABSTRACT FROM AUTHOR]

Aligned-Images Sum-set Inequalities are used in this work to study the Generalized Degrees of Freedom (GDoF) of the symmetric layered multi-hop interference channel under the robust assumption that the channel state information at the transmitters (CSIT) is limited to finite precision. First, the sum-GDoF value is characterized for the $2\times 2\times 2$ setting that is comprised of 2 sources, 2 relays, and 2 destinations. It is shown that the sum-GDoF does not improve even if perfect CSIT is allowed in the first hop, as long as the CSIT in the second hop is limited to finite precision. The sum GDoF characterization is then generalized to the $2\times 2\times \cdots \times 2$ setting that is comprised of $L$ hops. Remarkably, for large $L$ , the sum-GDoF value approaches that of the one-hop broadcast channel that is obtained by full cooperation among the two transmitters of the last hop, with finite precision CSIT. Previous studies of multi-hop interference networks either identified sophisticated GDoF optimal schemes under perfect CSIT, such as aligned interference neutralization and network diagonalization, that are powerful in theory but too fragile to be practical, or studied robust achievable schemes like classical amplify/decode/compress-and-forward without claims of information-theoretic optimality. In contrast, under finite precision CSIT, we show that the benefits of fragile schemes are lost, while a combination of classical random coding schemes that are simpler and much more robust, namely a rate-splitting between decode-and-forward and amplify-and-forward, is shown to be GDoF optimal. As such, this work represents another step towards bridging the gap between theory (optimality) and practice (robustness) with the aid of Aligned-Images Sum-set Inequalities. [ABSTRACT FROM AUTHOR]

We find the degrees of freedom of a network with S source nodes, R relay nodes, and D destination nodes, with random time-varying/frequency-selective channel coefficients and global channel knowledge at all nodes. We allow full-duplex operation at all nodes, as well as causal noise-free feedback of all received signals to all source and relay nodes. An outer bound to the capacity region of this network is obtained. Combining the outer bound with previous interference alignment based achievability results, we conclude that the techniques of relays, feedback, full-duplex operation and noisy cooperation do not increase the degrees of freedom of interference and X networks. As a second contribution, we show that for a network with K full-duplex nodes and K(K - 1) independent messages with one message from every node to each of the other K - 1 nodes, the total degrees of freedom are bounded above and below K(K - 1)/(2K - 2) and K(K - 1)/(2K - 3), respectively. Index Terms--Capacity, degrees of freedom, interference alignment, interference channel, multiple-input multiple-output (MIMO), X channel.

In wireless relay networks, noise at the relays can be correlated possibly due to common interference or noise propagation from preceding hops. A parallel relay network with noise correlation is considered in this network. For the relay strategy of amplify and forward (AF), the optimal rate maximizing relay gains when correlation knowledge is available at the relays are determined. Interestingly, it is shown that, on average, noise correlation is beneficial regardless of whether the relays know the noise covariance matrix. However, the knowledge of correlation can greatly improve the performance. Typically, the performance improvement from correlation knowledge increases with the relay power and the number of relays. With perfect correlation knowledge the system is capable of canceling interference if the number of interferers is less than the number of relays. For a two-hop multiple-access parallel network, closed-form expressions for the maximum sum rate and the optimal relay strategy are determined. Relay optimization for networks with three hops is also considered. Based on the result of two-hop network with noise correlation, an iterative algorithm is proposed for solving the relay optimization problem for three-hop networks. Index Terms--Amplify and forward (AF), common interference, multiple hop, noise correlation, parallel relay network, relay optimization.

An interference alignment example is constructed for the deterministic channel model of the K-user interference channel. The deterministic channel example is then translated into the Gaussian setting, creating the first known example of a fully connected Gaussian K-user interference network with single antenna nodes, real, nonzero and constant channel coefficients, and no propagation delays where the degrees of freedom outerbound is achieved. An analogy is drawn between the propagation delay based interference alignment examples and the deterministic channel model which also allows similar constructions for the two-user X channel as well. Index Terms--Capacity, degrees of freedom, interference alignment, interference channel, X channel.

In order to discover larger networks and parameter regimes where sharp Generalized Degrees of Freedom (GDoF) characterizations may be found based on the optimality of robust schemes for interference and broadcast networks with channel state information at the transmitters (CSIT) limited to finite precision, we explore the impact of secrecy constraints. In the absence of secrecy constraints, the largest such parameter regime for $K$ user interference networks is the CTIN regime (so named for the Convexity of the GDoF region achieved by Treating Interference as Noise (TIN)) originally discovered by Yi and Caire, whose optimality was established by Chan et al. For the corresponding broadcast networks the largest regime is the SLS (Simple Layered Superposition) regime discovered by Davoodi and Jafar, but only for small networks with $K\leq 3$ users. By including secrecy constraints, we identify much larger regimes, the STIN regime and the SLS regime, where GDoF are fully characterized for arbitrary number of users under finite precision CSIT, for interference networks and broadcast networks, respectively. The optimal achievable scheme in both cases is based on TIN along with power control and jamming. Proofs of optimality rely on a combination of secrecy constraints and Aligned Images sum-set inequalities. [ABSTRACT FROM AUTHOR]

The increasing prevalence of massive datasets makes the outsourcing of storage and computation tasks to distributed servers a necessity. This raises a number of concerns regarding the security and integrity of stored information, the privacy of accessing desired information, the communication overhead of distributed systems, the latency, reliability, and complexity of distributed computing, and privacy in distributed training and learning systems. Recent breakthroughs from coding, communication, and information-theoretic perspectives have opened up exciting new research avenues for these topics. There are many theoretical and practical open problems. This Special Issue is dedicated to communication theory, coding theory, information theory, signal processing, and networking aspects of privacy in information retrieval, privacy in coded computing over distributed servers, and privacy in distributed learning. [ABSTRACT FROM AUTHOR]

Most of our lives are conducted in the cyberspace. The human notion of privacy translates into a cyber notion of privacy on many functions that take place in the cyberspace. This article focuses on three such functions: how to privately retrieve information from cyberspace (privacy in information retrieval), how to privately leverage large-scale distributed/parallel processing (privacy in distributed computing), and how to learn/train machine learning models from private data spread across multiple users (privacy in distributed (federated) learning). The article motivates each privacy setting, describes the problem formulation, summarizes breakthrough results in the history of each problem, and gives recent results and discusses some of the major ideas that emerged in each field. In addition, the cross-cutting techniques and interconnections between the three topics are discussed along with a set of open problems and challenges. [ABSTRACT FROM AUTHOR]

We present a condition for 1/2 degree of freedom for each user in constant $K$ -user single-antenna interference channels. This condition is sufficient for all and necessary for almost all channel matrices. Moreover, it applies to all channel topologies, i.e., to fully-connected channels as well as channels that have individual links absent, reflected by corresponding zeros in the channel matrix. Moreover, it captures the essence of interference alignment by virtue of being expressed in terms of a generic injectivity condition that guarantees separability of signal and interference. Finally, we provide codebook constructions achieving 1/2 degree of freedom for each user for all channel matrices satisfying the condition we identified. [ABSTRACT FROM AUTHOR]

For the fully connected K user wireless interference channel where the channel coefficients are time-varying and are drawn from a continuous distribution, the sum capacity is characterized as C(SNR) = K/2 log(SNR) + o(log(SNR)). Thus, the K user time-varying interference channel almost surely has K/2 degrees of freedom. Achievability is based on the idea of interference alignment. Examples are also provided of fully connected K user interference channels with constant (not time-varying) coefficients where the capacity is exactly achieved by interference alignment at all SNR values. Index Terms--Capacity, degrees of freedom, interference alignment, interference channel, multiple-input-multiple-output (MIMO), multiplexing.

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 [less than or equal to] [[eta].sup.*.sub.X] [less than or equal to] 4/3. If M > 1 and channel matrices are nondegenerate then the precise degrees of freedom [[eta].sup.*.sub.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 X 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. Index Terms--Capacity, cognitive radio, degrees of freedom, dirty paper coding, interference alignment, multiple-input multiple-output (MIMO), zero forcing.

The ergodic capacity of a fading vector channel with multiple transmit antennas and a single receive antenna is explored. Perfect channel information is assumed to be available at the receiver while the transmitter has only partial knowledge of the direction of the user's channel vector based on quantized feedback. We present necessary and sufficient conditions for the optimality of beamforming in such systems. The conditions are applicable to all quantized feedback scenarios regardless of the channel distribution, number of transmit antennas, number of quantization vectors or transmit power. The optimality conditions are closely related to the iteration conditions of the Lloyd algorithm, revealing an interesting link between the optimality of beamforming and the optimality of the vector quantizers. Using the conditions, we prove the capacity optimality of beamforming for several quantized feedback scenarios such as the antenna-selection scheme. We also point out examples of quantized feedback scenarios where beamforming is not optimal. We find that for the independent identically distributed Rayleigh fading channel with more than a single bit of quantized feedback, there is no capacity benefit from increasing the number of antennas beyond the number of quantization vectors. Extensions of the necessary and sufficient optimality condition to the multiple-input multiple-output case are also provided. Index Terms--Beamforming, ergodic capacity, multiple-input, multiple-output (MIMO) channels, partial/limited feedback, quantized feedback.

In this paper, we consider multihop multiple access (MAC) and broadcast channels (BC) where communication takes place with the assistance of relays that amplify and forward (AF) their received signals. For a two-hop parallel AF relay MAC, assuming a sum power constraint across all relays we characterize optimal relay amplification factors and the resulting optimal rate regions. We find the maximum sum rate and the maximum rate for each user in closed form and express the optimal rate pair ([R.sub.1], [R.sub.2]) that maximizes [[micro].sub.1] R.sub.1] + [[micro].sub.2] [R.sub.2] as the solution of a pair of simultaneous equations. We find that the parallel AF relay MAC with total transmit power of the two users [P.sub.1] + [P.sub.2] = P and total relay power [P.sub.R] is the dual of the parallel AF relay BC where the MAC source nodes become the BC destination nodes, the MAC destination node becomes the BC source node, the dual BC source transmit power is [P.sub.R] and the total transmit power of the AF relays is P. The duality means that the rate region of the AF relay MAC with a sum power constraint P on the transmitters is the same as that of the dual BC. The duality relationship is found to be useful in characterizing the rate region of the AF relay BC as the union of MAC rate regions. The duality is established for distributed multiple antenna AF relay nodes and multiple (more than 2) hops as well. Index Terms--Amplify-and-forward (AF), broadcast, capacity, duality, multiple-input multiple-output (MIMO), multiple access.

In this correspondence, we show that the exact number of spatial degrees of freedom (DOF) for a two user nondegenerate (full rank channel matrices) multiple-input-multiple-output (MIMO) Gaussian interference channel with [M.sub.1],[M.sub.2] antennas at transmitters 1, 2 and [N.sub.1]\[N.sub.2] antennas at the corresponding receivers, and perfect channel knowledge at all transmitters and receivers, is min{[M.sub.1] + [M.sub.2] + [N.sub.2], max([M.sub.1], [N.sub.2]). max([M.sub.2], [N.sub.1]) }. A constructive achievability proof shows that zero forcing is sufficient to achieve all the available DOF on the two user MIMO interference channel. We also show through an example of a share-and-transmit scheme how the gains of transmitter cooperation may be entirely offset by the cost of enabling that cooperation so that the available DOF are not increased. Index Terms--Broadcast, degrees of freedom (DOF), interference, multiple--input--multiple--output (MIMO), multiple access, zero forcing.

The optimality of transmit beamforming for a multiple antenna system with partial/limited feedback is investigated and a single general necessary and sufficient condition for beamforming to achieve ergodic capacity is derived. The condition obtained is universal--applicable to all partial/limited feedback scenarios in all ergodic fading channel distributions regardless of the number of transmit/ receive antennas or transmit power, Using the universal condition we explore the optimality of beamforming for the quantized mean feedback scheme, which unifies previous results for the separate cases of mean feedback and quantized feedback. Numerical results are provided to complement the analysis. Index Terms--Beamforming, ergodic capacity, multiple-input-- multipl-eoutput (MIMO) channels, partial feedback, quantized feedback, quantized mean feedback.

We investigate the performance degradation of basic modulation schemes in a rapidly time-varying channel using a first-order autoregressive channel model. Various performance metrics are used to indicate the relative advantages of each modulation scheme. We find that noncoherent frequency-shift keying (FSK) is suitable for operating at very high mobility and high signal-to-noise ratio, ideal for some military applications. We then propose a partially coherent detector for FSK and differential phase-shift keying that exploits partial channel knowledge to enable the receiver to operate effectively in both fast and slow fading. The maximum-likelihood rule obtained for the partially coherent FSK turns out to be a linear combination of coherent and noncoherent detection rules. Results demonstrate that significant performance improvement can be achieved over the best of coherent and noncoherent FSK detection. The detector is robust to estimation errors present in the channel statistics. We also propose a few adaptive schemes that employ various combinations of modulation schemes to increase the robustness of the system in fast fading. Index Terms--Fast fading, interblock adaptation, intrablock adaptation, partially coherent detection, rapidly time-varying channel.

In this correspondence, we identify the common underlying form of the capacity expression that is applicable to both cases where causal or noncausal side information is made available to the transmitter. A genie-aided outerbound is developed that states that when a genie provides n bits of side information to a receiver the resulting capacity improvement cannot be more than n bits. Using the genie-bound we are able to bound the relative capacity advantage of noncausal side information over causal side information for both the single user point-to-point channel as well as the multiple-access channel (MAC) with independent side information at the transmitters. Applications of these capacity bounds are demonstrated through examples of random access channels. Interestingly, the capacity results indicate that the excessive MAC layer overheads common in present wireless systems may be avoided through coding across multiple-access blocks. It is also shown that even one bit of side information at the transmitter can result in unbounded capacity improvement. Index Terms--Capacity, causal, feedback, multiple access, mutual information, noncausal, side information.

In this correspondence, we determine the feedback capacity region of a two user Gaussian multiple-access channel (MAC) with multiple antennas at the base station and a single antenna at each user. The vector MAC with a single antenna at the base station and multiple antennas at each user is shown to be equivalent to a scalar MAC. We also determine the capacity enhancement due to feedback at high signal-to-noise ratio (SNR) for the scalar and vector MAC for any number of users. Extensions of the high SNR results to the vector broadcast channel are also provided. Index Terms--Broadcast, capacity, feedback, multiple access, multiple antennas.

We show that for highly mobile ad hoc networks, the benefits of mobility are overshadowed by the cost of mobility in terms of the increased channel uncertainty and network homogeneity. We assume a block-fading channel model with jointly isotropic fading. We allow relays which can transmit and receive simultaneously. Under fairly general assumptions for the users' channel fades and additive noise distributions we show that increasing the number of transmit antennas M at any node beyond the channel coherence time [T.sub.c] (measured in units of channel uses) does not affect the capacity region of the ad hoc network. For a fast-fading (coherence time [T.sub.c] [less than or equal to] M) homogeneour network, we determine the exact capacity region of the ad hoc network for any partition of the nodes into source, destination, and relay nodes. The optimal strategy is such that only one pair of source--destination nodes is active at a time while all the other nodes are inactive. There is no benefit from relaying and at high signal-to-noise ratio (SNR) the total throughput grows at most double-logarithmically with the number of nodes. Even for the case of slow fading, where the channel variations are slow enough that the receiver can track the channel perfectly, the inability of the transmitter to track the network topology limits the total throughput growth rate to no more than logarithmic in the number of nodes. Spatial correlation is shown to enhance the capacity region of the Rayleigh-fading ad hoc network. Index Terms--Ad hoc networks, capacity, fading, mobility, relay.

In this correspondence, we consider the problem of maximizing sum rate of a multiple-antenna Gaussian broadcast channel (BC). It was recently found that dirty-paper coding is capacity achieving for this channel. In order to achieve capacity, the optimal transmission policy (i.e., the optimal transmit covariance structure) given the channel conditions and power constraint must be found. However, obtaining the optimal transmission policy when employing dirty-paper coding is a computationally complex nonconvex problem. We use duality to transform this problem into a well-structured convex multiple-access channel (MAC) problem. We exploit the structure of this problem and derive simple and fast iterative algorithms that provide the optimum transmission policies for the MAC, which can easily be mapped to the optimal BC policies. Index Terms--Broadcast channel, dirty-paper coding, duality, multiple-access channel (MAC), multiple-input multiple-output (MIMO), systems.

We propose a scalar upper bound on the capacity region of the isotropic fading vector broadcast channel in terms of the capacity region of a scalar fading broadcast channel. The scalar upper bound is applicable to the broad class of isotropic fading broadcast channels regardless of the distribution of the users' channel magnitudes, the distribution of the additive noise experienced by each user, or the amount of channel knowledge available at the receiver. Using this upper bound, we prove the optimality of the Alamouti scheme in a broadcast setting, extend the recent results on the capacity of nondegraded, fading scalar broadcast channels to nondegraded fading vector broadcast channels, and determine the capacity region of a fading vector Gaussian broadcast channel with channel magnitude feedback. We also provide an example of a Rayleigh-fading broadcast channel with no channel state information available to the receiver (CSIR), where the bound on the capacity region obtained by a naive application of the scalar upper bound is provably loose, because it fails to account for the additional loss in degrees of freedom due to lack of channel knowledge at the receiver. A tighter upper bound is obtained by separately accounting for the loss in degrees of freedom due to lack of CSIR before applying the scalar upper bound. Index Terms--Broadcast channel, channel capacity, channel state information, degrees of freedom, fading channels, multiple antennas.