On first-order phase transition in microcanonical and canonical non-extensive systems

Two examples of Microcanonical Potts models, 2-dimensional nearest neighbor and mean field, are considered via exact enumeration of states and analytical asymptotic methods. In the interval of energies corresponding to a first order phase transition, both of these models exhibit a convex dip in the entropy vs energy plot and a region with negative specific heat within the dip. It is observed that in the nearest neighbor model the dip flattens and disappears as the lattice size grows, while in the mean field model the dip persists even in the limit of an infinite system. If formal transitions from microcanonical to canonical ensembles and back are performed for an infinite but non-extensive system, the convex dip in the microcanonical entropy plot disappears.Comment: 10 pages, 8 figure

Finite-Dimensional Bicomplex Hilbert Spaces

This paper is a detailed study of finite-dimensional modules defined on bicomplex numbers. A number of results are proved on bicomplex square matrices, linear operators, orthogonal bases, self-adjoint operators and Hilbert spaces, including the spectral decomposition theorem. Applications to concepts relevant to quantum mechanics, like the evolution operator, are pointed out.Comment: 21 page

History of the Department of Mathematics of the University of Kansas, 1886-1970

Francis Huntington Snow, in 1866, was the first member of the faculty to teach mathematics in The University of Kansas. From this auspicious beginning, mathematics developed into one of the major departments in the University; its staff has contributed many of the most distinguished and best loved members of the faculty. It is thus altogether fitting that someone should write a history of the Department of Mathematics and its staff. Furthermore, almost no information about the Department is available at the present time. Although Sterling's Quarter-Centennial History of The University of Kansas, 1866-1891, Robert Taft's Across the.Years on Mount Dread and The Years on Mount Dread, and Ryder's Snow of Kansas contain partial histories of the University, they yield almost no information about the Department of Mathematics or its mathematicians. Griffin's The University of Kansas: A History, published in 1974, is the University's official history; it contains more information about mathematics but not a complete history of the Department. The pioneers in the early years struggled against overwhelming obstacles and built a distinguished university. This History of the Department of Mathematics of the University of Kansas presents an account of their struggles and of their successes and failures, especially in the field of mathematics. I have written it as a tribute to the determination, the courage, the perseverance, and the memory of those who preceded us, and I hope that it will be also an inspiration and a challenge to those who follow