How to keep the arduino uno up on serial connections?
The reason for the reset is that Arduino Uno boards use the DTR signal line as a reset input, for automatic "rebooting" of the board for loading fresh code from the Arduino IDE.

One solution suggested (for Arduino Uno boards) is to connect a 10 μF capacitor between the Reset pin and Ground. This should prevent the auto-reset.

Another, more permanent solution is to cut the RESET-EN trace on the Uno board with a sharp knife. Re-enabling serial port based reset involves creating a solder bridge on the cut trace to reconnect it.

With either of the above solutions you will need to press the Reset button each time just before the IDE begins uploading code to the board. Getting this manual reset timing right may need a bit of practice.