Preface. I. Spline Functions. Total Positivity of the Spline Kernel and its Applications; B. Bojanov. Total Positivity Properties of LB-Splines; T. Lyche, L.L. Schumaker. Total Positivity and Splines; K. Mørken. Totally Positive Systems and Interpolation by Subspaces of Generalized Splines; M. Sommer. H. Strauss. II. Matrix Theory. Preservation of Total Nonnegativity under the Hadamard Product and Related Topics; J. Garloff, D.G. Wagner. Vertex Implications for Totally Nonnegative under the Hadamard Product and Related Topics; J. Garloff. On Factorizations of Totally Positive Matrices; J. Garloff. On Factorizations of Totally Positive Matrices; M. Gasca, J.M. Peña. III. Geometric Modeling. Total Positivity and Optimal Bases; J.M. Carnicer, J.M. Peña. Total Positivity and the Shape of Curves; T.N.T. Goodman. Tchebycheff Curves; M.L. Mazure, H. Pottmann. IV. Probability and Mathematical Biology. Analytical and Probabilistic Methods for Totally Positive Matrices; G.S. Goodman. Some Inequalities of Total Positivity in Pure and Applied Mathematics; S. Karlin, Z. Ziegler. New Directions and Problems of Multivariate Total Positivity Motivated by Genetic Recombination Probabilities; S. Karlin. V. Approximation Theory. Tchebycheff Spaces and Total Positivity; J.M. Carnicer, J.M. Peña. Cebysev and Weak Cebysev Systems; R.A. Zalik. VI. Complex Analysis. Zero Sets of Entire Fourier Transforms of Multiply Positive Density Functions; O.M. Katkova. VII. Statistics. Total Nonnegativity of Moment Matrices and its Application to E- Optimal Designs in Polynomial Regression; B. Heiligers. Analysis of Regression Dependence in 2 xn Bivariate Distributions and Some Applications in Contingency Tables; S. Kasala, M.B. Rao. VIII. Real Analysis. Total Positivity and Nonlinear Analysis; R.-Q. Jia. Solutions of Certain Functional Equations; S. Karlin. Alternance and Snakes in the Nonlinear Finite Moment Problem; K. Tatalian. IX. Combinatorics. The Applications of Total Positivity to Combinatorics, and Conversely; F. Brenti. X. Integral Equations. Spectral Properties of Totally Positive Kernels and Matrices; A. Pinkus. Index.

This volume contains both invited lectures and contributed talks presented at the meeting on Total Positivity and its Applications held at the guest house of the University of Zaragoza in Jaca, Spain, during the week of September 26-30, 1994. There were present at the meeting almost fifty researchers from fourteen countries. Their interest in thesubject of Total Positivity made for a stimulating and fruitful exchange of scientific information. Interest to participate in the meeting exceeded our expectations. Regrettably, budgetary constraints forced us to restriet the number of attendees. Professor S. Karlin, of Stanford University, who planned to attend the meeting had to cancel his participation at the last moment. Nonetheless, his almost universal spiritual presence energized and inspired all of us in Jaca. More than anyone, he influenced the content, style and quality of the presentations given at the meeting. Every article in these Proceedings (except some by Karlin hirnself) references his influential treatise Total Positivity, Volume I, Stanford University Press, 1968. Since its appearance, this book has intrigued and inspired the minds of many researchers (one of us, in his formative years, read the galley proofs and the other of us first doubted its value but then later became its totally committed disciple). All of us present at the meeting encourage Professor Karlin to return to the task of completing the anxiously awaited Volume 11 of Total Positivity.