Let (φ, ψ) be an (m, n)-valued pair of maps φ, ψ : X Y , where φ is an m-valued map and ψ is n-valued, on connected finite polyhedra. A point x ∈ X is a coincidence point of φ and ψ if φ(x) ∩ ψ(x) = ∅. We define a Nielsen coincidence number N (φ : ψ) which is a lower bound for the number of coincidence points of all (m, n)-valued pairs of maps homotopic to (φ, ψ). We calculate N (φ : ψ) for all (m, n)-valued pairs of maps of the circle and show that N (φ : ψ) is a sharp lower bound in that setting. Specifically, if φ is of degree a and ψ of degree b, thenwhere < m, n > is the greatest common divisor of m and n. In order to carry out the calculation, we obtain results, of independent interest, for nvalued maps of compact connected Lie groups that relate the Nielsen fixed point number of Helga Schirmer to the Nielsen root number of Michael Brown.
Here, we report on the development and evaluation of novel unobstructing magnetic microactuators for maintaining the patency of implantable ventricular catheters used in hydrocephalus application. The treatment of hydrocephalus requires chronic implantation of a shunt system to divert excess cerebrospinal fluid from the brain. These shunt systems suffer from a high failure rate (>40%) within the first year of implantation, often due to biological accumulation. Previously, we have shown that magnetic microactuators can be used to remove biological blockage. The new cantilever-based magnetic microactuator presented in this paper improves upon the previous torsional design using a bimorph to induce a postrelease out-of-plane deflection that will prevent the device from occluding the pore at rest. The mechanical evaluations (i.e., postrelease deflection, static and dynamic responses) of fabricated devices are reported and compared with theoretical values.
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