Linear and Nonlinear Trees: Multiplicity Lists of Symmetric Matrices

Let A be a real symmetric matrix whose graph is a tree, T. If T is a linear tree (meaning all vertices with degree 3 or larger lie on the same induced path), then we can use a ”Linear Superposition Principle” to determine all possible multiplicities of eigenvalues of A. If T is a nonlinear tree, we must use other ad hoc methods. I utilize these methods to compute all possible multiplicity lists of trees on 12 vertices, and augment an existing multiplicities database. This database allows us to examine of the effects that the structure of tree can have on a multiplicity list. Then, I investigate the enumeration of linear and nonlinear trees, and examine the ratio of nonlinear trees to total trees

ACCF/SCAI/SVMB/SIR/ASITN 2007 clinical expert consensus document on carotid stenting - A report of the American College of Cardiology Foundation Task Force on Clinical Expert Consensus Documents (ACCF/SCAI/SVMB/SIR/ASITN Clinical Expert Consensus Document

Stroke is considered as the third leading cause of death in the US, behind heart disease and cancer, there are 1 million stroke-related deaths each year, including 500,000 new strokes, 200,000 recurrent strokes, and 240,000 transient ischemic attacks (TIA). The patients with temporary retinal or hemispheric neurological deficits should be screened for extracranial carotid artery disease. Non invasive imaging technique is a useful tool to assess carotid stenosis severity and guide treatment. Carotid duplex is the most widely available and least expensive non invasive imaging procedure. Cardiovascular risk factor modification is recommended to limit the progression of atherosclerosis and disease clinical events and antiplatelet therapy is recommended for symptomatic patients