Soil surface textures, Cumberland County, New Jersey /

"October 1986."

On the size of the fibers of spectral maps induced by semialgebraic embeddings

Let S(M) be the ring of (continuous) semialgebraic functions on a semialgebraic set M and S*(M) its subring of bounded semialgebraic functions. In this work we compute the size of the fibers of the spectral maps Spec(j)1:Spec(S(N))→Spec(S(M)) and Spec(j)2:Spec(S*(N))→Spec(S*(M)) induced by the inclusion j:N M of a semialgebraic subset N of M. The ring S(M) can be understood as the localization of S*(M) at the multiplicative subset WM of those bounded semialgebraic functions on M with empty zero set. This provides a natural inclusion iM:Spec(S(M)) Spec(S*(M)) that reduces both problems above to an analysis of the fibers of the spectral map Spec(j)2:Spec(S*(N))→Spec(S*(M)). If we denote Z:=ClSpec(S*(M))(M N), it holds that the restriction map Spec(j)2|:Spec(S*(N)) Spec(j)2-1(Z)→Spec(S*(M)) Z is a homeomorphism. Our problem concentrates on the computation of the size of the fibers of Spec(j)2 at the points of Z. The size of the fibers of prime ideals "close" to the complement Y:=M N provides valuable information concerning how N is immersed inside M. If N is dense in M, the map Spec(j)2 is surjective and the generic fiber of a prime ideal p∈Z contains infinitely many elements. However, finite fibers may also appear and we provide a criterium to decide when the fiber Spec(j)2-1(p) is a finite set for p∈Z. If such is the case, our procedure allows us to compute the size s of Spec(j)2-1(p). If in addition N is locally compact and M is pure dimensional, s coincides with the number of minimal prime ideals contained in p. © 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

Acidic substrata potential map : [Monmouth County, New Jersey] /

"This map estimates the potential of encountering a sulfide bearing geologic substratum beneath the soil."

Acidic substrata potential map : [Middlesex County, New Jersey] /

"This map estimates the potential of encountering a sulfide bearing geologic substratum beneath the soil."

Acidic substrata potential map : [Mercer County, New Jersey] /

"This map estimates the potential of encountering a sulfide bearing geologic substratum beneath the soil."

Soil surface textures, Salem County, New Jersey /

"October 1986."

Soil surface textures, Salem County, New Jersey /

"October 1986."

FIRM, flood insurance rate map and street index, borough of Carteret, New Jersey, Middlesex County /

Relief shown by spot heights.

Edison National Historic Site, West Orange, New Jersey. May 1957.

Scale ca. 1:1,100.

A plan of the county of Fairfax on Potomack River the middle of which is in 39⁰, 12ʹ No. latitude.

Also covers Arlington and Loudoun counties.

Captn. Willoby. Newton's including deed for 6421 acrer of land in the County of Fairfax : [Virginia] : Deed drawn /

Shows the parcel bounded by Bull Run Creek to the south, Cub Run to the west, and Little Rocky Run to the east in Fairfax County, Virginia and names of landowners.

Map of Alexandria, Fairfax, Prince William, Stafford, and portions of the adjacent county's [sic].

Scale ca. 1:126,720.

Map of 1st. distrt. Campbell Co. Georgia : south of the Cherokee boundy. line /

Shows local topography, roads, and names of some residents.

Plan of the entrance of Barataria with a projected battery proposed by Lt. Coll. G.T. Ross. No. 9 Rapport May 3d 1813 /

Relief shown pictorially. Depths shown by soundings.

Map of Milwaukee County and plans of Pewaukee, Oconomowoc, Wauwatosa, Waukesha /

Includes index.