In 1990, Cook, Kannan and Schrijver proved that the split closure (the 1st 1-branch split closure) of a polyhedron is again a polyhedron. They also gave an example of a 3-dimensional mixed-integer polytope whose 1-branch split rank is infinite. We generalize this example to a family of high-dimensional polytopes and present a closed-form description of the kth 1-branch split closure of these po...

According to Alexander's law (AL), the slow phase velocity of nystagmus of vestibular origin is dependent on horizontal position, with lower velocity when gaze is directed in the slow compared with the fast phase direction. Adaptive changes in the velocity-to-position neural integrator are thought to cause AL. Although these changes have been described for the horizontal neural integrator, nyst...

We study the dual complexes of boundary divisors in log resolutions of compactifications of algebraic varieties and show that the homotopy types of these complexes are independent of all choices. Inspired by recent developments in nonarchimedean geometry, we consider relations between these boundary complexes and weight filtrations on singular cohomology and cohomology with compact supports, an...

In November 2014 Alexander Grothendieck passed away at the age of 86. There is no doubt that he was one of the greatest and most innovative mathematicians of the 20th century. After a bitter childhood, his meteoric ascent started in the Cartan Seminar in Paris, it led to a breakthrough while he worked in São Paulo, and to the Fields Medal. He introduced numerous new concepts and techniques, whi...

In this article, we introduced and explore the properties of two sets of functions: weighted pseudo periodic functions of class r, and weighted Stepanov-like pseudo periodic functions of class r. We show the existence and uniqueness of weighted pseudo periodic solution of class r that are solutions to neutral functional differential equations. Other applications to partial differential equation...