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This paper presents a reconfigurable intelligent surface (RIS) design and simulation aimed at enhancing beamforming and beam steering capabilities for 5G and 6G mobile communications. The proposed design introduces a 2- bit unit cell design, having four distinct phase states that can be tuned by a single varactor diode. This configuration has a 5x5 array and provides efficient operation at 23.8 GHz within the 5G New Radio (NR) frequency range 2 (FR2). The proposed RIS design demonstrates unique beam steering capabilities ranging from −60∘ to 60∘ in the azimuth plane which is crucial for extending coverage into the mm-wave coverage. The performance of the RIS is simulated using the CST 3D electromagnetic simulator, focusing on radar cross section (RCS) pattern for optimization. The simulation results reveal effective beam steering capabilities ranging from -10° to -60° and 10° to 60°, with a minimum scan loss of approximately 3 dB. The proposed RIS exhibits the high angular reciprocity that handles the incident waves up to 110∘ at an oblique 60∘ angle. 

We prove that the higher rho invariant is a homomorphism from the structure group of a compact manifold to the K-group of certain geometric C*-algebra. In particular, we apply this result to show that the structure group is infinitely generated for a class of manifolds.