Problem 1. Infinite Products and Coproducts in Ab. We have seen that finite products and coproducts agree in Ab. However, the same is not true for infinite products and coproducts. Let I be a set and let {Ai}i∈I be a family of abelian groups, each equal to some fixed group A. (a) Show that the set AI := HomSet(I, A) is an abelian group. Furthermore, show that we can think of this group as the i...

Even though biological legs operate amazingly springlike, they clearly deviate from a perfect spring. For instance, experimental data of human hopping and running suggest an increase in leg length and a decrease in leg stiffness between touchdown and takeoff. Here, this variability of human gait is considered as a fundamental system property and introduced via non-constant leg parameters to a 1...

Zero stiffness structures have the remarkable ability to undergo large elastic deformations without requiring external work. Several equivalent descriptions exist, namely i) continuous equilibrium, ii) constant potential energy, iii) neutral stability and iv) zero stiffness. Each perspective on zero stiffness provides different methods of analysis and design. This paper reviews the concept of z...

In this paper, for the purpose of improving the ride comfort of a passenger car, the use of an air-cell on top of the conventional polyurethane foam seat is proposed. The air-cell can be modeled as a variable spring with the change of its internal air pressure. Therefore, for different human sizes and road disturbances, the adjustment of the air-cell can make the optimal contact pressure betwee...

A simple and accurate experimental method is described for determining the effective cantilever spring constant and the detector sensitivity of atomic force microscopy cantilevers on which a colloidal particle is attached. By attaching large (approximately 85 microm diameter) latex particles at various positions along the V-shaped cantilevers, we demonstrate how the normal and lateral spring co...

Using quantitative phase microscopy, we have discovered a quadratic relationship between the radius R and the thickness t of helical ribbons that form spontaneously in multicomponent cholesterol-surfactant mixtures. These helical ribbons may serve as mesoscopic springs to measure or to exert forces on nanoscale biological objects. The spring constants of these helices depend on their submicrosc...

In the present study, numerical simulations of the ow in a three-dimensional axisymmet-ric poppet valve are conducted using a vortex method. The uids are assumed inviscid and incompressible. The vibration of the poppet, supported by a spring and a dash pot, occurs in the axial direction in some range of parameters. The numerical simulations are carried out by varying systematically the paramete...

Supplementary Method: How to Construct the Constraint Matrix Here, we provide the detailed procedure to construct the constraint matrix, because it is essential process in component mode synthesis that is employed in our coarse-graining method. For straightforward illustration, instead of considering complex structure such as protein structure, we restrict ourselves to one-dimensional spring sy...