Calculus I: The first derivative of pseudoisotopy theory

Let P be the (stable, smooth) pseudoisotopy space of the space X. For any map Y: YX of spaces, we identify the homotopy type of the fiber of P(f): P(f) P(f) in a stable range, roughly twice the connectivity of the map YX. We establish some language for discussing and manipulating such stable-range relative calculations for any homotopy functor. The theorem about P has a corollary about Waldhausen's A. Dedicated to Alexander GrothendieckResearch supported in part by NSF grant DMS-8806444 and by a Sloan Fellowship. Research at MSRI supported in part by NSF grant DMS-8505550.     

Service stage Petri net models with product form solution

A class of service stage Petri net models whose equilibrium probabilities satisfy local balance equations is presented. Examples of their applicability include bus oriented multiprocessor interconnections, the classical dining philosophers problem and the alternating bit protocol. The natural topological space for embedding the state transition lattice for this class of SSPN is shown to be a multidimensional toroidal manifold.     

Presentations of the first homotopy groups of the unitary groups

We describe explicit presentations of all stable and the first nonstable homotopy groups of the unitary groups. In particular, for each n 2 we supply n homotopic maps that each represent the (n - 1)!-th power of a suitable generator of _2n SU(n) _n!. The product of these n commuting maps is the constant map to the identity matrix.     

Gomory integer programs

 The set of all group relaxations of an integer program contains certain special members called Gomory relaxations. A family of integer programs with a fixed coefficient matrix and cost vector but varying right hand sides is a Gomory family if every program in the family can be solved by one of its Gomory relaxations. In this paper, we characterize Gomory families. Every TDI system gives a Gomory family, and we construct Gomory families from matrices whose columns form a Hilbert basis for the cone they generate. The existence of Gomory families is related to the Hilbert covering problems that arose from the conjectures of Seb?. Connections to commutative algebra are outlined at the end. Received: May 17, 2001 / Accepted: February 7, 2002 Published online: April 24, 2003 RID="⋆" ID="⋆" Research partially supported by NSF grant DMS-0100141.     

Flow Past a Swept Wing with a Compliant Surface: Stabilizing the Attachment-Line Boundary Layer

Many aquatic species such as dolphins and whales have fins, which can be modeled as swept wings. Some of these fins, such as the dorsal fin of a dolphin, are semi-rigid and therefore can be modeled as a rigid swept wing with a compliant surface. An understanding of the hydrodynamics of the flow past swept compliant surfaces is of great interest for understanding potential drag reduction mechanisms, especially since swept wings are widely used in hydrodynamic and aerodynamic design. In this paper, the flow past a swept wing with a compliant surface is modeled by an attachment-line boundary layer flow, which is an exact similarity solution of the Navier–Stokes equations, flowing past a compliant surface modeled as an elastic plate. The hydrodynamic stability of the coupled problem is studied using a new numerical framework based on exterior algebra. The basic instability of the attachment line boundary layer on a rigid surface is a traveling wave instability that propagates along the attachment line, and numerical results show that the compliance results in a substantial reduction in the instability region. Moreover, the results show that, although the flow-field is three-dimensional, the qualitative nature of the instability suppression is very similar to the qualitative reduction of instability of the two-dimensional Tollmien–Schlichting modes in the classical boundary-layer flow past a compliant surface.     

Relationship Between Professional Development,Teachers' Instructional Practices,and the Achievement of Students in Science and Mathematics

The purpose of this study was to examine the relationship between different types of professional development, teachers' instructional practices, and the achievement of students in science and mathematics. The types of professional development studied included immersion, examining practice, curriculum implementation, curriculum development, and collaborative work. Data regarding teachers' instructional practices and the amount of professional development were collected using teacher surveys. Ninety‐four middle school science teachers and 104 middle school mathematics teachers participated in the study. Student achievement was measured using eighth grade state science and mathematics achievement test data. Regression analyses suggested that for both science and mathematics teachers, examining practice and curriculum development were significantly related to the use of standards‐based instructional practices. Only curriculum development for mathematics teachers was significantly related to student achievement. Implications of results for the professional development of science and mathematics teachers are discussed.     

A contribution to shape theory for the ising model at low temperature

Summary We consider low temperature limits of Gibbs states of the ferromagnetic nearest-neighbour Ising Hamiltonian in the positive orthant of the lattice ^d ,d=1, 2,..., under a negative boundary condition and a small positive external fieldh that decreases linearly with the temperatureT. It is shown that positive and negative spins are separated by a staircase-shaped random boundary. Its explicit distribution is computed in the case that the ratio =h/T exceeds some positive ₀. For < ₀, our results do not rule out infinite negative areas.