The Krein-Milman theorem in operator convexity

We generalize the Krein-Milman theorem to the setting of matrix convex sets of Effros-Winkler, extending the work of Farenick-Morenz on compact C-convex sets of complex matrices and the matrix state spaces of C-algebras. An essential ingredient is to prove the non-commutative analogue of the fact that a compact convex set may be thought of as the state space of the space of continuous affine functions on .

     

Nonparametric tests for bounds on the derivative of a regression function

We consider two tests of the null hypothesis that the k-th derivative of a regression function is uniformly bounded by a specified constant. These tests can be used to study the shape of the regression function. For instance, we can test for convexity of the regression function by setting k=2 and the constant equal to zero. Our tests are based on k-th order divided difference of the observations. The asymptotic distribution and efficacies of these tests are computed and simulation results presented.Research supported by Natural Sciences and Engineering Research Council of Canada Grant OGP0007969.Research supported by National Science Foundation Grant DMS-9306738.     

On the monotonicity of the power functions of tests based on traces of multivariate beta matrices

It is shown that for the MANOVA problem the power function of the test based on the trace of a multivariate beta matrix is monotonically increasing in each noncentrality parameter provided that the cutoff point is not too large. This result is also true for the problem of testing independence of two sets of variates.     

On isoperimetric problems for domains with partly known boundaries

In the present paper, we discuss the isoperimetric problems for domains with partly known boundaries, i.e., the problem of determining a domain that minimizes the capacity functional in the class of plain double-connected domains having the same fixed area and outer boundary. The formulas for capacity variations obtained in this paper allows us to formulate necessary conditions.It is proved that the convexity of the fixed outer boundary implies the convexity of the inner boundary corresponding to an optimal domain. Then, we discuss the case where the fixed part of the boundary is a square.Further, we consider similar problems with more complicated functionals. We introduce the concept of a minimal function in the class of equimeasurable functions. This concept allows us to unify the approach to all of these problems. At the end, we produce a hypothesis that, if proved, would enable us to characterize the shape of the optimal domains in the isoperimetric problems mentioned above.The author wishes to express his appreciation to Dr. K. A. Lurie for his help and unceasing attention.     

Synthesis of a first thiophene containing analog of the HIV protease inhibitor nelfinavir

A stereoselective and efficient preparation of a thiophene containing intermediate 2 from ethyl 3-thienyl propenoate 4 as the core of new possible HIV protease inhibitors is described. The chiral intermediate has been successfully used for the preparation of the analog 1 of the potent HIV inhibitor nelfinavir.     

On approximately convex functions

A real valued function defined on a real interval is called -convex if it satisfies

The main results of the paper offer various characterizations for -convexity. One of the main results states that is -convex for some positive and if and only if can be decomposed into the sum of a convex function, a function with bounded supremum norm, and a function with bounded Lipschitz-modulus. In the special case , the results reduce to that of Hyers, Ulam, and Green obtained in 1952 concerning the so-called -convexity.

     

An example in holomorphic fixed point theory

If is the open unit ball in the Cartesian product furnished with the -norm , where and , then a holomorphic self-mapping of has a fixed point if and only if for some

     

The Forcing Convexity Number of a Graph

For two vertices u and v of a connected graph G, the set I(u,v) consists of all those vertices lying on a u-v geodesic in G. For a set S of vertices of G, the union of all sets I(u,v) for u, v S is denoted by I(S). A set S is a convex set if I(S) = S. The convexity number con(G) of G is the maximum cardinality of a proper convex set of G. A convex set S in G with |S| = con(G) is called a maximum convex set. A subset T of a maximum convex set S of a connected graph G is called a forcing subset for S if S is the unique maximum convex set containing T. The forcing convexity number f(S, con) of S is the minimum cardinality among the forcing subsets for S, and the forcing convexity number f(G, con) of G is the minimum forcing convexity number among all maximum convex sets of G. The forcing convexity numbers of several classes of graphs are presented, including complete bipartite graphs, trees, and cycles. For every graph G, f(G, con) con(G). It is shown that every pair a, b of integers with 0 a b and b is realizable as the forcing convexity number and convexity number, respectively, of some connected graph. The forcing convexity number of the Cartesian product of H × K ₂ for a nontrivial connected graph H is studied.