Generalized Weighted Morrey Estimates for Marcinkiewicz Integrals with Rough Kernel Associated with Schrödinger Operator and Their Commutators

Let L =-?+V(x) be a Schr?dinger operator, where ? is the Laplacian on ■~n,while nonnegative potential V(x) belonging to the reverse H?lder class. The aim of this paper is to give generalized weighted Morrey estimates for the boundedness of Marcinkiewicz integrals with rough kernel associated with Schr?dinger operator and their commutators.Moreover, the boundedness of the commutator operators formed by BMO functions and Marcinkiewicz integrals with rough kernel associated with Schr?dinger operators is discussed on the generalized weighted Morrey spaces. As its special cases, the corresponding results of Marcinkiewicz integrals with rough kernel associated with Schr?dinger operator and their commutators have been deduced, respectively. Also, Marcinkiewicz integral operators, rough Hardy-Littlewood(H-L for short) maximal operators, Bochner-Riesz means and parametric Marcinkiewicz integral operators which satisfy the conditions of our main results can be considered as some examples.     

Observation of Reform Teaching in Undergraduate Level Mathematics and Science Courses

This paper reports on initial results from an ongoing evaluation study of a National Science Foundation project to implement reform‐oriented teaching practices in college science and mathematics courses. The purpose of this study was to determine what elements of reform teaching are being utilized by college faculty members teaching undergraduate science and mathematics courses, including a qualitative estimate of the frequency with which they are used. Participating instructors attended summer institutes that modeled reform‐based practices and fostered reflection on current issues in science, mathematics, and technological literacy for K‐16 teaching, with an explicit emphasis on the importance of creating the best possible learning experience for prospective K‐12 science and mathematics teachers. Utilizing a unique classroom observation protocol (the Oregon‐Teacher Observation Protocol) and interviews, the authors (a) conclude that some reform‐oriented teaching strategies are evident in undergraduate mathematics and science instruction and (b) suggest areas in which additional support and feedback are needed in order for higher education faculty members to adopt reform‐based instructional methodology.     

A queuing system with time varying rates

We obtain the queue length distribution of a multiple-server queuing system with time varying arrival and service rates when these rates are high.     

Anion Effects on Sodium Ion and Acid Molecule Adduction to Protein Ions in Electrospray Ionization Mass Spectrometry

Gaseous protein–metal ion and protein–molecule complexes can be readily formed by electrospray ionization (ESI) from aqueous solutions containing proteins and millimolar concentrations of sodium salts of various anions. The extent of sodium and acid molecule adduction to multiply charged protein ions is inversely related and depends strongly on the proton affinity (PA) of the anion, with extensive sodium adduction occurring for anions with PA values greater than ~300 kcal·mol^–1 and extensive acid molecule adduction occurring for anions with PA values less than 315 kcal·mol^–1. The role of the anion on the extent of sodium and acid molecule adduction does not directly follow the Hofmeister series, suggesting that direct protein–ion interactions may not play a significant role in the observed effect of anions on protein structure in solution. These results indicate that salts with anions that have low PA values may be useful solution-phase additives to minimize nonspecific metal ion adduction in ESI experiments designed to identify specific protein-metal ion interactions.