Steven J. Cox

The rate at which energy decays in a damped String

The energy in a string subject to positive viscous damping is known to decay exponentially in time. Under the assumption that the damping is of bounded variation, we identify the best rate of decay with the supremum of the real part of the spectrum of the infinitesimal generator of the underlying semigroup. We analyze the spectrum of this nonselfadjoint operator in some detail. Our bounds on its real eigenvalues and asymptotic form of its large eigenvalues translate into criteria for over/underdamping and a proof that the decay rate achieves its (negative) minimum over those dampings whose total variation does not exceed a prescribed value.

On the optimal design of columns against buckling

The authors establish existence, derive necessary conditions, infer regularity, and construct and test an algorithm for the maximization of a column’s Euler buckling load under a variety of boundary conditions over a general class of admissible designs. It is proven that symmetric clamped-clamped columns possess a positive first eigenfunction and a symmetric rearrangement is introduced that does not decrease the column’s buckling load. The necessary conditions, expressed in the language of Clarke’s generalized gradient [10], subsume those proposed by Olhoff and Rasmussen [25], Masur [22], and Seiranian [34]. The work of [25], [22], and [34] sought to correct the necessary conditions of Tadjbakhsh and Keller [37], who had not foreseen the presence of a multiple least eigenvalue. This remedy has been hampered by Tadjbakhsh and Keller’s miscalculation of the buckling loads of their clamped-clamped and clamped-hinged columns. This issue is resolved in the appendix.In the numerical treatment of the associated ...

Extremal eigenvalue problems for composite membranes, II

Given an open bounded connected set Ω ⊂RN and a prescribed amount of two homogeneous materials of different density, for smallk we characterize the distribution of the two materials in Ω that extremizes thekth eigenvalue of the resulting clamped membrane. We show that these extremizers vary continuously with the proportion of the two constituents. The characterization of the extremizers in terms of level sets of associated eigenfunctions provides geometric information on their respective interfaces. Each of these results generalizes toN dimensions the now classical one-dimensional work of M. G. Krein.

The Shape of the Tallest Column

The height at which an unloaded column will buckle under its own weight is the fourth root of the least eigenvalue of a certain Sturm--Liouville operator. We show that the operator associated with the column proposed by Keller and Niordson [J. Math. Mech., 16 (1966), pp. 433--446] does not possess a discrete spectrum. This calls into question their formal use of perturbation theory, so we consider a class of designs that permits a tapered summit yet still guarantees a discrete spectrum. Within this class we prove that the least eigenvalue increases when one replaces a design with its decreasing rearrangement. This leads to a very simple proof of the existence of a tallest column.

Morphologically accurate reduced order modeling of spiking neurons

Accurately simulating neurons with realistic morphological structure and synaptic inputs requires the solution of large systems of nonlinear ordinary differential equations. We apply model reduction techniques to recover the complete nonlinear voltage dynamics of a neuron using a system of much lower dimension. Using a proper orthogonal decomposition, we build a reduced-order system from salient snapshots of the full system output, thus reducing the number of state variables. A discrete empirical interpolation method is then used to reduce the complexity of the nonlinear term to be proportional to the number of reduced variables. Together these two techniques allow for up to two orders of magnitude dimension reduction without sacrificing the spatially-distributed input structure, with an associated order of magnitude speed-up in simulation time. We demonstrate that both nonlinear spiking behavior and subthreshold response of realistic cells are accurately captured by these low-dimensional models.

Achieving Arbitrarily Large Decay in the Damped Wave Equation

We exhibit a sequence of viscous dampings for the fixed string that yields arbitrarily fast attenuation of any and all initial disturbances. The limit case produces extinction of all solutions in finite time.

Perturbing the critically damped wave equation

We consider the wave equation with viscous damping. The equation is said to be critically damped when the damping is that value for which the spectral abscissa of the associated wave operator is minimized within the class of constant dampings. The critically damped wave operator possesses a nonsemisimple eigenvalue. We present a detailed study of the splitting of this eigenvalue under bounded perturbations of the damping and subsequently show that the critical choice is a local minimizes of the spectral abscissa over lines in the class of all bounded dampings.

The two phase drum with the deepest bass note

Given an open bounded connected set δ ⊂ R2 and a prescribed amount of two homogeneous materials of different density we characterize that distribution which minimizes the least eigenvalue of the associated clamped drum. We establish geometric conditions on δ under which the interface separating the two materials is an analytic Jordan curve. We bound the length of this interface and construct and test an algorithm for its calculation in the case of square δ. Our numerical results depict the dependence of this minimum least eigenvalue on the volume fractions of the two phases and suggest possible candidates for the two phase drum with the least second eigenvalue.

One Can Hear the Composition of a String: Experiments with an Inverse Eigenvalue Problem

To what extent do the vibrations of a mechanical system reveal its composition? Despite innumerable applications and mathematical elegance, this question often slips through those cracks that separate courses in mechanics, differential equations, and linear algebra. We address this omission by detailing a classical finite dimensional example: the use of frequencies of vibration to recover positions and masses of beads vibrating on a string. First we derive the equations of motion, then compare the eigenvalues of the resulting linearized model against vibration data measured from our laboratorys monochord. More challenging is the recovery of masses and positions of the beads from spectral data, a problem for which a variety of elegant algorithms exist. After presenting one such method based on orthogonal polynomials in a manner suitable for advanced undergraduates, we confirm its efficacy through physical experiment. We encourage readers to conduct their own explorations using the numerous data sets we provide.

Recovering Quasi-Active Properties of Dendritic Neurons from Dual Potential Recordings

We develop the theory and accompanying algorithm for the recovery of a dendritic neurons cytoplasmic resistivity, membrane capacitance, leakage conductance, and two maximal channel conductances from weighted averages of simultaneous recordings of somatic and dendritic potential following a somatic current stimulus. We test our results on two model systems with distinct, though prescribed, channel kinetics and branching patterns.