By John P. D’Angelo, Mihai Putinar (auth.), Mihai Putinar, Seth Sullivant (eds.)

Recent advances in either the idea and implementation of computational algebraic geometry have resulted in new, extraordinary purposes to a number of fields of research.

The articles during this quantity spotlight a number of those purposes and supply introductory fabric for themes lined within the IMA workshops on "Optimization and keep an eye on" and "Applications in Biology, Dynamics, and information" held through the IMA yr on purposes of Algebraic Geometry. The articles relating to optimization and keep an eye on concentrate on burgeoning use of semidefinite programming and second matrix concepts in computational actual algebraic geometry. the hot path in the direction of a scientific research of non-commutative actual algebraic geometry is easily represented within the quantity. different articles offer an summary of ways computational algebra turns out to be useful for research of contingency tables, reconstruction of phylogenetic bushes, and in platforms biology. The contributions accumulated during this quantity are obtainable to non-experts, self-contained and informative; they fast circulation in the direction of innovative learn in those components, and supply a wealth of open difficulties for destiny research.

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Where we take Xl == xi, X2 == x2 is a symmetric noncommutative rational expression. 2. We return to convexity checker command and illustrate it on F( (a, b,r), (x, y)) :== - (y + a*xb)(r + b*xb)-l(y + b*xa) + a*xa. 1) where x == z", y == y*. Here we are viewing F as a function of two classes of variables (see Section 4). An application of the command NCConvexityRegion[F, {x, y}] outputs the list {-2 (r + b*xb)-l, 0, 0, O}. 36 MAURICIO C. DE OLIVEIRA ET AL. This output has the meaning that whenever A, B, R are fixed matrices, the function F is "x, y-matrix concave" on the domain of matrices X, and Y QA,B,R:== {(X,Y): (R+B*XB)-l >- O}.

This set of inequalities while not usually convex in X, Yare convex in the new variables W == X -1 and Z == Y -1, since DCK F X and DG K Fy are linear in Wand Z and X - y-l == W- 1 - Z has second derivative 2W- 1 HW-l HW- 1 which is non negative in H for each W- 1 == X >- o. These inequalities are also equivalent to LMIs which we do not write down. 9. Classical RAG extended to free-* algebras. At this point one might think of the emerging area of free *- semi-algebraic geometry as having two main paths.

4. 2 the upper 2 x 2 block of £"'((X) is negative definite if and only if I - L(X) >- 0 if and only if X is in the component of 0 of the domain of r. Given that the upper 2 x 2 block of L"'((X) is negative definite, by the LDL* (Cholesky) factorization, 0 >- L; (X) is negative definite if and only if II >- r(X). D 6. Ideas behind some proofs and the convexity checker algorithm. 2, the results from Section 4 on polynomials in two classes of variables, and many of the results on rational functions exposited in the previous section begin, just as in the case of everywhere convex polynomials, with the observation that matrix convexity of a noncommutative rational function on a noncommutative convex domain is equivalent to its noncommutative second directional derivative being matrix positive.