He believed that by paying Mexico's debt to America would let him have the settlements that he wanted to get the Rio Grande, pay the debt settlement of Mexico in exchange for his conditions, and by California and Mexico as well. Polk thought that since there is a tension between America and Mexico it was best to get in and open some agreements.
Even though the devastation caused by the atomic bomb was tremendous, and had killed hundreds of thousands of Japanese citizens, the Emperor refused to back down. Surrender in the Japanese culture is shunned very harshly as it made one cowardly.