A General Global Optimization Approach for Solving Location Problems in the Plane

We propose a general approach for constructing bounds required for the “Big Triangle Small Triangle” (BTST) method for the solution of planar location problems. Optimization problems, which constitute a sum of individual functions, each a function of the Euclidean distance to a demand point, are analyzed and solved. These bounds are based on expressing each of the individual functions in the sum as a difference between two convex functions of the distance, which is not the same as convex functions of the location. Computational experiments with nine different location problems demonstrated the effectiveness of the proposed procedure.     

Chattering limit for a model of harvesting in a rapidly changing environment

An example of harvesting in an environment that changes rapidly and frequently is examined. The changes are unpredictable, yet their statistical distribution is known. The parameters, however, can be measured on line, and used in the control policy. We show that then chattering systems adequately serve as nominal systems, optimal solutions of which provide near optimal solutions for the system with rapid changes.This research was supported by a grant from the Basic Research Fund, the Israel Academy of Science and Humanities. Zvi Artstein is the incumbent of the Hettie H. Heineman Professorial Chair in Mathematics.     

Location of an Obnoxious Route

This problem deals with the location of a route or path through a set of given points in order to maximize the smallest weighted distance from the given points to the route. Applications may include the planning of pipelines carrying noxious material, and also certain problems in robotics. The first algorithm finds a non-linear path by iteratively solving network minimal-cut problems. A second algorithm solves the case where the route is restricted to be linear.     

Averaged local field intensities in composite films

The photophysical properties of molecules adsorbed in composite films (e.g. surface island films) depend on the local electromagnetic field within the film. The ratio between the average field intensity 〈|E|²〉 in the film and the intensity |E_I|² associated with the incident field is a measure of the electromagnetic contribution to the surface influence on molecular photophysical phenomena. This ratio depends on the film composition and morphology, on the dielectric properties of the pure components making the film and on the frequency, direction and polarization of the incident radiation. Calculations of this ratio as a function of these parameters for several models of composite films are presented. Image interactions and retardation effects as well as radiative damping and finite size contributions to the dielectric response of the films are taken into account. In addition, an estimate of the field inhomogeneity within the film is obtained by calculating also the ratio 〈|E|²〉_shell/|E_I|² associated with the field in thin shells surrounding the dielectric particles which constitute the film.     

A calculus for set-valued maps and set-valued evolution equations

A definition of differentiability of a set-valued map is offered. As derivatives, which are called directives in the set-valued setting, unions of affine maps are used; these are called multiaffines. A multiaffine is a directive if it is a first-order approximation of the set-valued map. One application is a necessary condition for maximin optimality of constrained decisions. A distance among multiaffines permits the development of set-valued evolution equations along the lines of ordinary differential equations in a vector space. The theory is displayed along with some comments on applications.Incumbent of the Hettie H. Heineman Professorial Chair in Mathematics.     

Thermal expansion of polycrystalline aggregates: I. Exact analysis

An exact relation is developed between the thermal expansion coefficient and the bulk modulus of statistically isotropic polycrystalline aggregates composed of crystals of hexagonal, tetragonal or trigonal symmetry. This relation is exploited to derive simple close bounds for the thermal expansion coefficient in terms of single crystal properties. Comparison of bounds to experimentally obtained expansion coefficients shows fair to very good agreement.     

Three-dimensional microfiber devices that mimic physiological environments to probe cell mechanics and signaling

Many physiological systems are regulated by cells that alter their behavior in response to changes in their biochemical and mechanical environment. These cells experience this dynamic environment through an endogenous biomaterial matrix that transmits mechanical force and permits chemical exchange with the surrounding tissue. As a result, in vitro systems that mimic three-dimensional, in vivo cellular environments can enable experiments that reveal the nuanced interplay between biomechanics and physiology. Here we report the development of a minimal-profile, three-dimensional (MP3D) experimental microdevice that confines cells to a single focal plane, while allowing the precise application of mechanical displacement to cells and concomitant access to the cell membrane for perfusion with biochemical agonists. The MP3D device--an ordered microfiber scaffold erected on glass--provides a cellular environment that induces physiological cell morphologies. Small manipulations of the scaffold's microfibers allow attached cells to be mechanically probed. Due to the scaffold's minimal height profile, MP3D devices confine cells to a single focal plane, facilitating observation with conventional epifluorescent microscopy. When examining fibroblasts within MP3D devices, we observed robust cellular calcium responses to both a chemical stimulus as well as mechanical displacement of the cell membrane. The observed response differed significantly from previously reported, mechanically-induced calcium responses in the same cell type. Our findings demonstrate a key link between environment, cell morphology, mechanics, and intracellular signal transduction. We anticipate that this device will broadly impact research in fields including biomaterials, tissue engineering, and biophysics.