Semi-inclusive triple differential multiplicity distributions of positively charged kaons have been measured over a wide range in rapidity and transverse mass for central collisions of $^{58}$Ni with $^{58}$Ni nuclei. The transverse mass ($m_t$) spectra have been studied as a function of rapidity at a beam energy 1.93 AGeV. The $m_t$ distributions of K^+ mesons are well described by a single Boltzmann-type function. The spectral slopes are similar to that of the protons indicating that rescattering plays a significant role in the propagation of the kaon. Multiplicity densities have been obtained as a function of rapidity by extrapolating the Boltzmann-type fits to the measured distributions over the remaining phase space. The total K^+ meson yield has been determined at beam energies of 1.06, 1.45, and 1.93 AGeV, and is presented in comparison to existing data. The low total yield indicates that the K^+ meson can not be explained within a hadro-chemical equilibrium scenario, therefore indicating that the yield does remain sensitive to effects related to its production processes such as the equation of state of nuclear matter and/or modifications to the K^+ dispersion relation.
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