Uniqueness of one-dimensional N\'eel wall profiles

We study the domain wall structure in thin uniaxial ferromagnetic films in the presence of an in-plane applied external field in the direction normal to the easy axis. Using the reduced one-dimensional thin film micromagnetic model, we analyze the critical points of the obtained non-local variational problem. We prove that the minimizer of the one-dimensional energy functional in the form of the N\'eel wall is the unique (up to translations) critical point of the energy among all monotone profiles with the same limiting behavior at infinity. Thus, we establish uniqueness of the one-dimensional monotone N\'eel wall profile in the considered setting. We also obtain some uniform estimates for general one-dimensional domain wall profiles.Comment: 18 page

Supporting GENP with Random Multipliers

We prove that standard Gaussian random multipliers are expected to stabilize numerically both Gaussian elimination with no pivoting and block Gaussian elimination. Our tests show similar results where we applied circulant random multipliers instead of Gaussian ones.Comment: 14 page

Biosensing with microcantilever-based sensors

Microcantilevers provide an ideal platform for biosensors. The micron-sized transducer brings several advantages, such as high sensitivity, small sample quantity for analysis, portability, implantable sensor devices, and the ability to be mass produced and integrated into standard microelectronic processing technologies like complementary metal oxide seminconductors (CMOS). The objective of this research is to investigate and develop modification methods of microcanitlevers for biosensing applications. Two microcantilever modification methods were investigated. They are self-assembly monolayer method and layer-by-layer method. A fundamental procedure for modification of microcantilevers using the layer-by-layer approach was developed for the first time in this research. These modification methods for microcantilevers provide practical ways for immobilization of recognition specifics, such as enzymes and antibodies, on the surface of the microcantilever. The modifications allow for detection of corresponding targets. In this research, the following results have been obtained: (1) Development of a glucose sensor using microcantilever with layer-by-layer nano assembly containing glucose oxidas. The sensor has a response time in the range of seconds. (2) Development of a hydrogen peroxides sensor using the microcantilever with layer-by-layer nano assembly containing hydrogen peroxides. The detection limit for this sensor is 10−9M. (3) Development of a sensor for detection of biowarfare agents. For the measurement of Tularemia, this sensor reached the detection limit of 103 organism/ml. (4) Development of a sensor for detection of chemical warfare agents with sensitivity of 10−7M for organophosphates. The results obtained from this research have demonstrated that the microcantilever-based biosensors can be developed for detection of various biomoleules or monitoring different processes. The glucose sensor developed in this research has great potential to be used as implantable glucose sensor for continuous blood glucose monitoring, which is critical in diabetes care. And the sensor for detection of biowarfare agents could be used for homeland security, which is one of the most important issues of the nation

Preconditioning For Matrix Computation

Preconditioning is a classical subject of numerical solution of linear systems of equations. The goal is to turn a linear system into another one which is easier to solve. The two central subjects of numerical matrix computations are LIN-SOLVE, that is, the solution of linear systems of equations and EIGEN-SOLVE, that is, the approximation of the eigenvalues and eigenvectors of a matrix. We focus on the former subject of LIN-SOLVE and show an application to EIGEN-SOLVE. We achieve our goal by applying randomized additive and multiplicative preconditioning. We facilitate the numerical solution by decreasing the condition of the coefficient matrix of the linear system, which enables reliable numerical solution of LIN-SOLVE. After the introduction in the Chapter 1 we recall the definitions and auxiliary results in Chapter 2. Then in Chapter 3 we precondition linear systems of equations solved at every iteration of the Inverse Power Method applied to EIGEN-SOLVE. These systems are ill conditioned, that is, have large condition numbers, and we decrease them by applying randomized additive preconditioning. This is our first subject. Our second subject is randomized multiplicative preconditioning for LIN-SOLVE. In this way we support application of GENP, that is, Gaussian elimination with no pivoting, and block Gaussian elimination. We prove that the proposed preconditioning methods are efficient when we apply Gaussian random matrices as preconditioners. We confirm these results with our extensive numerical tests. The tests also show that the same methods work as efficiently on the average when we use random structured, in particular circulant, preconditioners instead, but we show both formally and experimentally that these preconditioners fail in the case of LIN-SOLVE for the unitary matrix of discreet Fourier transform, for which Gaussian preconditioners work efficiently

A Liouville-type theorem for higher order elliptic systems

AbstractWe prove there are no positive radial solutions for higher order elliptic system{(−Δ)mu=vp(−Δ)mv=uqinRN if 1p+1+1q+1>1−2mN. We also show there are no positive solutions to the system under the additional assumption that max(2(p+1)pq−1,2(q+1)pq−1)⩾N−2mm. The proof in the radial case uses Rellichʼs identity and the proof in the general case relies on growth estimates of the spherical average of the solution