Leveraging Galois Theory and Computational Results in the Search for Legendre Pairs

With applications spanning myriad disciplines, Hadamard matrices have tremendous utility. Infinitely many have been discovered, but there is no general proof of their existence. Proving the Hadamard conjecture would provide a significant technological edge. Hadamard matrices are difficult to construct in general; however, they can be created directly from Legendre pairs (LPs). While LPs are not trivial to find, a breakthrough in this area would be pivotal in the quest to prove the Hadamard conjecture. Most recent efforts rely upon refined search algorithms, and we seek to either develop a better such algorithm or discover an altogether new theoretical construction. By leveraging Galois theory, we are able to gain greater insight into the power spectral densities (PSDs) of LPs. We also develop a new theorem and conjecture about LP compressions. Finally, we provide a new standard for recording and reporting LPs which will benefit future efforts in this area

Addressing the Gap of Informal Science Field Experiences in Science Methods Courses

Preservice teachers in science education courses do not generally experience informal, authentic science education practices in their fieldwork. This study used the Scholarship of Teaching and Learning to frame a study that integrated informal science teaching experiences with a university teacher education science methods course. The results showed improvement in participant preservice teachers\u27 ability to teach science content, their self-confidence in teaching science, and their ability to connect what they learned in their methods course in a real-life setting

Development of risk maps to minimize uranium exposures in the Navajo Churchrock mining district

© 2009 deLemos et al; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution Licens

Phase I study of pegylated liposomal doxorubicin and the multidrug-resistance modulator, valspodar

Valspodar, a P-glycoprotein modulator, affects pharmacokinetics of doxorubicin when administered in combination, resulting in doxorubicin dose reduction. In animal models, valspodar has minimal interaction with pegylated liposomal doxorubicin (PEG-LD). To determine any pharmacokinetic interaction in humans, we designed a study to determine maximum tolerated dose, dose-limiting toxicity (DLT), and pharmacokinetics of total doxorubicin, in PEG-LD and valspodar combination therapy in patients with advanced malignancies. Patients received PEG-LD 20–25 mg m−2 intravenously over 1 h for cycle one. In subsequent 2-week cycles, valspodar was administered as 72 h continuous intravenous infusion with PEG-LD beginning at 8 mg m−2 and escalated in an accelerated titration design to 25 mg m−2. Pharmacokinetic data were collected with and without valspodar. A total of 14 patients completed at least two cycles of therapy. No DLTs were observed in six patients treated at the highest level of PEG-LD 25 mg m−2. The most common toxicities were fatigue, nausea, vomiting, mucositis, palmar plantar erythrodysesthesia, diarrhoea, and ataxia. Partial responses were observed in patients with breast and ovarian carcinoma. The mean (range) total doxorubicin clearance decreased from 27 (10–73) ml h−1 m−2 in cycle 1 to 18 (3–37) ml h−1 m−2 with the addition of valspodar in cycle 2 (P=0.009). Treatment with PEG-LD 25 mg m−2 in combination with valspodar results in a moderate prolongation of total doxorubicin clearance and half-life but did not increase the toxicity of this agent

Symmetry Groups for Linear Programming Relaxations of Orthogonal Array Problems

Integer linear programs arise in many situations, and solving such problems can be computationally demanding. One way to solve them more efficiently is by exploiting the symmetry within their formulation. This paper proves that the symmetry group for the linear programming relaxation of 2-level orthogonal array problems of strength 2 is a particular semidirect product

The Linear Programming Relaxation Permutation Symmetry Group of an Orthogonal Array Defining Integer Linear Program

There is always a natural embedding of $S_{s}\wr S_{k}$ into the linear programming (LP) relaxation permutation symmetry group of an orthogonal array integer linear programming (ILP) formulation with equality constraints. The point of this paper is to prove that in the $2$ -level, strength- $1$ case the LP relaxation permutation symmetry group of this formulation is isomorphic to $S_{2}\wr S_{k}$ for all $k$ , and in the $2$ -level, strength- $2$ case it is isomorphic to $S_{2}^{k}\rtimes S_{k+1}$ for $k\geqslant 4$ . The strength- $2$ result reveals previously unknown permutation symmetries that cannot be captured by the natural embedding of $S_{2}\wr S_{k}$ . We also conjecture a complete characterization of the LP relaxation permutation symmetry group of the ILP formulation