The U.S. dollar stayed firm in the mid-161 yen range Tuesday morning in Tokyo after hitting a fresh 37-year high overnight, supported by rising U.S. Treasury yields that reinforced the prospect of a persistent wide interest rate gap between the United States and Japan. At noon, the dollar fetched 161.60-63 yen compared with 161.44-54 yen in New York after hitting 161.72 yen, its highest since December 1986, and 161.01-03 yen in Tokyo at 5 p.m. Monday. The euro was quoted at $1.0735-0736 and 173.48-53 yen against $1.0735-0745 and 173.40-50 yen in New York and $1.0754-0756 and 173.16-20 yen in T…