In this paper, by definition of EXPONENTIAL MAP of the Lie groups the concept of EXPONENTIAL MAP of generalized Lie groups is introduced. This has a powerful generalization to generalized Lie groups which takes each line through the origin to an order product of some one-parameter subgroup. We show that the EXPONENTIAL MAP is a C¥- MAP. Also, we prove some important properties of the EXPONENTIAL MAP for generalized Lie groups. Under the identification, it is shown that the derivative of the EXPONENTIAL MAP is the identity MAP. One of the most powerful applications of these EXPONENTIAL MAPs is to define generalized adjoint representation of a top space, so we show that this representation is a C¥- MAP. Finally, invariant forms are introduced on a generalized Lie group. We prove that every left invariant K-form are introduced on a generalized Lie group T with the finite number of identity elements is C¥. At the end of this paper, for compact connected generalized Lie group T with the finite number of identity elements and dimension n, we show that every left invariant n-form on T is right invariant n-form.

We introduce a new wrapped EXPONENTIAL distribution named transmuted wrapped EXPONENTIAL (TWE) distribution, for the modeling of circular datasets by using the Transmutation Rank-MAP method. This method is employed for the first time for a wrapped distribution with this study. The introduced distribution is more flexible than traditional wrapped EXPONENTIAL distribution. The paper provides the explicit form of important distributional properties of the introduced distribution such as expectation, median, moments, characteristic function, quantile function, hazard rate function and stress-strength reliability. Ré nyi and Shannon entropies are also obtained. The statistical inference problem for the TWE distribution is investigated using maximum likelihood, least squares and weighted least squares and comparative numerical study results are presented. Furthermore, we present a real dataset analysis.

Welch & Peers (1963) used a root-information prior to obtain posterior probabilities for a scalar parameter EXPONENTIAL model and showed that these Bayes probabilities had the confidence property to second order asymptotically. An important undercurrent of this indicates that the constant information reparameterization provides location model structure, for which the confidence property was and is well known. This paper examines the role of the scalar-parameter EXPONENTIAL model for obtaining approximate probabilities and approximate confidence levels, and then addresses the extension for the vector-parameter EXPONENTIAL model.

Communication in the presence of a priori known interference at the encoder has gained great interest because of its many practical applications. In this paper, additive EXPONENTIAL noise channel with additive EXPONENTIAL interference (AENC-AEI) known noncausally at the transmitter is introduced as a new variant of such communication scenarios. First, it is shown that the additive Gaussian channel with a priori known interference at the encoder when the transmitter suffers from a fast-varying phase noise can be modeled by the AENC-AEI. Then, capacity bounds for this channel under a non-negativity constraint as well as a mean value constraint on input are derived. Finally, it is shown both analytically and numerically that the upper and lower bounds coincide at high signal to noise ratios (SNRs), and therefore, the capacity of the AENC-AEI at high SNRs is obtained. Interestingly, this high SNR-capacity has a simple closed-form expression and is independent of the interference mean, analogous to its Gaussian counterpart.

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