We present a constructive description of minimal reductions with a given reduction number. This description has interesting consequences on the minimal reduction number, the big reduction number, and the core of an ideal. In particular, it helps solve a conjecture of Vasconcelos on the relationship between reduction numbers and initial ideals.

Fibrotic diseases such as scleroderma, severe chronic asthma, pulmonary fibrosis, and cardiac fibrosis kill tens of thousands of people each year in the U.S. alone. Growing evidence suggests that in fibrotic lesions, a subset of blood monocytes enters the tissue and differentiates into fibroblast-like cells called fibrocytes, causing tissue dysfunction. We previously found that a plasma protein...

We use the p-adic local Langlands correspondence for GL2(Qp) to explicitly compute the reduction modulo p of certain 2-dimensional crystalline representations of small slope, and give applications to modular forms.

This report is based on my lecture at the Langlands conference in Princeton in 1996 and the series of lectures I gave at the semestre Hecke in Paris in 2000. In putting the notes for these lectures in order, it was my original intention to give a survey of the activities in the study of the reduction of Shimura varieties. However, I realized very soon that this task was far beyond my capabiliti...

The three art gallery problems Vertex Guard, Edge Guard and Point Guard are known to be NP -hard [8]. Approximation algorithms for Vertex Guard and Edge Guard with a logarithmic ratio were proposed in [7]. We prove that for each of these problems, there exists a constant > 0, such that no polynomial time algorithm can guarantee an approximation ratio of 1 + unless P = NP . We obtain our results...

PURPOSE To compare the efficacy and safety of open reduction and internal fixation through ilioinguinal approach and Stoppa approach for the treatment of displaced acetabular fractures. METHODS Case-controlled trials (CCTs) published from January 2010 to August 2015 that compared the ilioinguinal approach and Stoppa approach in the management of displaced acetabular fractures were retrieved f...