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How can I connect Activiti Eclipse to a database management system and then link that database with user interfaces (web form)?
I don't know what "Activiti Eclipse" is. Eclipse is a Java IDE and it does have a process designer plugin for Activiti, but other than that, the two are not related.
If you mean Activiti, do you mean Activiti embedded in Alfresco or do you mean standalone Activiti? Or do you mean Alfresco Activiti, which is the commercially supported standalone Activiti product? There are some helpers in Alfresco Activiti that allow you to design "entities" that map to database tables.