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Multi-waveband observations of thirty quasars and active galaxies with flat radio spectra (referred to here as FRSQ's) have been conducted to determine their radio to infrared and X-ray to gamma-ray properties and the relationships between them. Specifically, the synchrotron self-Compton (SSC) process is examined as a likely common radiation mechanism. Very Long Baseline Interferometry is used in conjunction with radio to submillimeter-wave observations to determine the parameters of the synchrotron spectrum. These parameters are used to predict the SSC X-ray flux densities. Seven of thirty sources have predicted Compton X-ray flux densities well above the observed flux densities obtained with the ROSAT satellite. It is concluded that these sources experience internal relativistic motion directed toward the observer's line of sight, which lowers the predicted Compton X-ray flux. Three of these seven sources are detected at gamma-ray frequencies. Model spectra show that the X-rays are consistent with the first order SSC process. The gamma-rays are not produced via second order self-Compton scattering, but are rather produced either by first order self-Compton scattering or by another process. A comparison of the ROSAT X-ray flux densities and those obtained earlier with the Einstein Observatory show that many FRSQ's are X-ray variables, with a factor of 2--3 between minimum and maximum brightness. Sources that were observed more than once with ROSAT also show variability over this shorter timescale. In several sources (e.g., 4C 39.25 and NRAO 140) the X-ray variability is associated with millimeter variability. Gamma-ray detections are also related to increases in the radio to millimeter-wave spectrum. Model spectra show that the relationship between the predicted high energy and radio to infrared spectra is sensitive to whether magnetic field or electron energy density dominates the variations. Implications for the SSC model and models that include scattering of external photons are discussed. Statistical analysis shows that the millimeter-wave and X-ray luminosities for the sample are strongly correlated, with a linear regression slope $\sim$ 0.6. The peak in the distribution of X-ray to millimeter spectral indices also indicates a strong connection between millimeter-wave and X-ray emission.
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