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Cognitive radio is an emerging technology that shows great promise to dramatically improve the efficiency of spectrum utilization. This paper considers a cognitive radio model, in which the secondary network is allowed to use the radio spectrum concurrently with primary users (PUs) provided that interference from the secondary users (SUs) to the PUs is constrained by certain thresholds. The weighted sum rate maximization problem is studied under interference power constraints and individual transmit power constraints, for a cognitive multiple access channel (C-MAC), in which each SU having a single transmit antenna communicates with the base station having multiple receive antennas. An iterative algorithm is developed to efficiently obtain the optimal solution of the weighted sum rate problem for the C-MAC. It is further shown that the proposed algorithm, although developed for single channel transmission, can be extended to the case of multiple channel transmission. Corroborating numerical examples illustrate the convergence behavior of the algorithm and present comparisons with other existing alternative algorithms.