By David Mumford

From the reports: "Although numerous textbooks on glossy algebraic geometry were released meanwhile, Mumford's "Volume I" is, including its predecessor the purple e-book of sorts and schemes, now as ahead of some of the most very good and profound primers of recent algebraic geometry. either books are only real classics!" Zentralblatt

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**Sample text**

A ∈ Rn1 ×n2 : A ≤ 1}◦ = {B ∈ Rn1 ×n2 : B ∗ ≤ 1}. Therefore, any two matrices A and B satisfy A, B ≤ A B ∗. 12) 1 1 where A F is the Frobenius norm, deﬁned as A F := (TrAT A) 2 = ( ij a2ij ) 2 . Both the operator norm and the nuclear norm have nice characterizations in terms of semideﬁnite programming. In particular, the operator norm A is the optimal solution of the primal-dual pair of semideﬁnite programs maximize Tr 2AT X12 subject to Tr X11 T X12 X 0, X12 = 1, X22 minimize subject to t tIn1 AT A tIn2 0.

D21n d22n d23n ... ⎤ d21n d22n ⎥ ⎥ ⎥ d23n ⎥ ⎥ .. ⎥ . ⎦ 0 is negative semideﬁnite on the subspace orthogonal to the vector e := (1, 1, . . , 1). Proof. We show only the necessity of the condition. , there are points xi ∈ Rk such that dij = xi − xj . Consider now the Gram matrix G of inner products ⎡ ⎢ ⎢ G := ⎢ ⎣ x1 , x1 x2 , x1 .. x1 , x2 x2 , x2 .. ... .. x1 , xn x2 , xn .. xn , x1 xn , x2 ... xn , xn ⎤ ⎥ ⎥ ⎥ = [x1 , . . , xn ]T [x1 , . . , xn ], ⎦ which is positive semideﬁnite by construction.

The Nevanlinna–Pick interpolation problem has many important applications in systems and control theory; see, for instance, [14] and [47] and the references therein. 5 Euclidean Distance Matrices Assume we are given a list of pairwise distances between a ﬁnite number of points. Under what conditions can the points be embedded in some ﬁnite-dimensional space and those distances be realized as the Euclidean metric between the embedded ✐ ✐ ✐ ✐ ✐ ✐ ✐ 38 main 2012/11/1 page 38 ✐ Chapter 2. Semideﬁnite Optimization points?