ANTIQUES ROADSHOW host Mark Walberg welcomes viewers to Providence, Rhode Island, where he and appraiser Chris Mitchell head for the Ocean State's storied coastline and onward to Rose Island Lighthouse near Newport for a look at Naval collectibles. Appraisers at the Rhode Island Convention Center encounter a fleet of admirable objects, including a valuable 1850s Massachusetts folk art weathervane, rescued from a demolition site; the 1786 diary of John Francis, son-in-law of John Brown, the prominent Providence merchant whose family name is carried by Brown University; and a rifle, powder horn and log book - attributed to Welcome Mathewson, one of the premier New England gunsmiths of the early 19th century - handed down through seven generations of a Rhode Island family, and valued at $100,000 to $120,000.