Converse symmetry and Intermediate energy values in rearrangement optimization problems

This paper discusses three rearrangement optimization problems where the energy functional is connected with the Dirichlet or Robin boundary value problems. First, we consider a simple model of Dirichlet type, derive a symmetry result, and prove an intermediate energy theorem. For this model, we show that if the optimal domain (or its complement) is a ball centered at the origin, then the original domain must be a ball. As for the intermediate energy theorem, we show that if $\alpha,\beta$ denote the optimal values of corresponding minimization and maximization problems, respectively, then every $\gamma$ in $(\alpha,\beta)$ is achieved by solving a max-min problem. Second, we investigate a similar symmetry problem for the Dirichlet problems where the energy functional is nonlinear. Finally, we show the existence and uniqueness of rearrangement minimization problems associated with the Robin problems. In addition, we shall obtain a symmetry and a related asymptotic result

A Landscape Analysis of Constraint Satisfaction Problems

We discuss an analysis of Constraint Satisfaction problems, such as Sphere Packing, K-SAT and Graph Coloring, in terms of an effective energy landscape. Several intriguing geometrical properties of the solution space become in this light familiar in terms of the well-studied ones of rugged (glassy) energy landscapes. A `benchmark' algorithm naturally suggested by this construction finds solutions in polynomial time up to a point beyond the `clustering' and in some cases even the `thermodynamic' transitions. This point has a simple geometric meaning and can be in principle determined with standard Statistical Mechanical methods, thus pushing the analytic bound up to which problems are guaranteed to be easy. We illustrate this for the graph three and four-coloring problem. For Packing problems the present discussion allows to better characterize the `J-point', proposed as a systematic definition of Random Close Packing, and to place it in the context of other theories of glasses.Comment: 17 pages, 69 citations, 12 figure

HIGH PERFORMANCE PIEZOELECTRIC MATERIALS AND DEVICES FOR MULTILAYER LOW TEMPERATURE CO-FIRED CERAMIC BASED MICROFLUIDIC SYSTEMS

The incorporation of active piezoelectric elements and fluidic components into micro-electromechanical systems (MEMS) is of great interest for the development of sensors, actuators, and integrated systems used in microfluidics. Low temperature cofired ceramics (LTCC), widely used as electronic packaging materials, offer the possibility of manufacturing highly integrated microfluidic systems with complex 3-D features and various co-firable functional materials in a multilayer module. It would be desirable to integrate high performance lead zirconate titanate (PZT) based ceramics into LTCC-based MEMS using modern thick film and 3-D packaging technologies. The challenges for fabricating functional LTCC/PZT devices are: 1) formulating piezoelectric compositions which have similar sintering conditions to LTCC materials; 2) reducing elemental inter-diffusion between the LTCC package and PZT materials in co-firing process; and 3) developing active piezoelectric layers with desirable electric properties. The goal of present work was to develop low temperature fired PZT-based materials and compatible processing methods which enable integration of piezoelectric elements with LTCC materials and production of high performance integrated multilayer devices for microfluidics. First, the low temperature sintering behavior of piezoelectric ceramics in the solid solution of Pb(Zr0.53,Ti0.47)O3-Sr(K0.25, Nb0.75)O3 (PZT-SKN) with sintering aids has been investigated. 1 wt% LiBiO2 + 1 wt% CuO fluxed PZT-SKN ceramics sintered at 900oC for 1 h exhibited desirable piezoelectric and dielectric properties with a reduction of sintering temperature by 350oC. Next, the fluxed PZT-SKN tapes were successfully laminated and co-fired with LTCC materials to build the hybrid multilayer structures. HL2000/PZT-SKN multilayer ceramics co-fired at 900oC for 0.5 h exhibited the optimal properties with high field d33 piezoelectric coefficient of 356 pm/V. A potential application of the developed LTCC/PZT-SKN multilayer ceramics as a microbalance was demonstrated. The final research focus was the fabrication of an HL2000/PZT-SKN multilayer piezoelectric micropump and the characterization of pumping performance. The measured maximum flow rate and backpressure were 450 μl/min and 1.4 kPa respectively. Use of different microchannel geometries has been studied to improve the pumping performance. It is believed that the high performance multilayer piezoelectric devices implemented in this work will enable the development of highly integrated LTCC-based microfluidic systems for many future applications