Four Projects in Applied Mathematics

该项目包括将数学应用于四个重要的物理问题：(1)制定从冷冻血液中去除甘油的最佳方案；(2)制定将一种形状弯曲和拉伸成另一种形状的最佳策略(例如，将金属板弯曲成弯曲的结构表面或在数字成像中变形)；(3)板上流动的水波模式的稳定性；以及(4)现代物理两体问题(即两个物体的运动理论，其运动产生波，波以有限的速度传播，影响系统中另一物体的运动)。必须在全血中加入甘油，以防止在冷冻时对细胞造成损害，并且必须在输血前将其去除。目前的清除战略需要太长的程序，不适合紧急情况或军事应用。利用数学建模(包括耦合流体流动、对流扩散和半透膜传输的模型)、优化理论和数值方法，该项目将扩大实验室测试的医用去除方案。最优变形的研究利用微分几何和弹性理论来获得最小应变变形的代价泛函。我们将用变分法证明可容许变形集(紧致超曲面之间的微分同胚)上存在极小值，并发展数值方法来逼近这些极小值。利用流体动力学模型和自由边界问题理论，证明了平板绕流产生的表面波的存在性和稳定性。此外，数学分析将与物理系统的数值模拟相辅相成。将推导出两体相互作用的数学模型，其中力以有限的速度传播。这种模型由双曲型偏微分方程组和常微分方程组组成，或者由微分方程组和时滞方程组成的混合方程组组成。将确定这两类模型的适定性和定性动力学，并将应用这类模型来理解声表面波传感器的同步现象。brbr该研究项目有两个主要目的：应用数学来增进对物理科学四个领域的理解，以及在这些重要的应用数学领域培养四名博士生。第一个项目在健康科学和医学方面有直接的应用。活细胞，例如血细胞，可以通过冷冻保存很长一段时间；然后，它们可以解冻用于输血。冷冻和解冻活细胞需要一项复杂的技术，包括在细胞内部添加名为冷冻保护剂的化学物质，以便在此过程中不会受到损害。在输血之前，必须去除这些化学物质。有方法可以做到这一点，但当前的方法需要很长时间才能实现。因此，冷冻血液不能用于紧急情况(例如，在军事冲突或自然灾害中)。冷冻保护剂的快速去除已被证明在单细胞水平上是可能的；研究的目的是设计(最佳)方法，能够用于医学上实际应用所需的更大规模的快速去除。第二个项目在材料科学中有应用。一块金属片要被弯曲和拉伸成所需的形状。所需的形状可以通过许多不同的变形过程来实现(例如，可以先拉伸薄板，然后弯曲，或者可以在一个方向上拉伸，同时在另一个方向上弯曲，以及许多其他可能性)。哪种方法使用的能量最少？这个问题和其他同类问题将用数学分析来解决。第三个项目将确定通常由流体(例如水)在板块上运动形成的表面波的稳定性。众所周知，面波是存在的，并且具有特殊的形状。此外，实验表明，当波受到干扰时，它们会迅速返回到那个特定的形状。但是，目前这一显著稳定性的机理尚不清楚。研究项目将确定观察到的稳定性的原因。第四个研究项目涉及物理和工程中的两体相互作用理论。应用声学中的一个实际例子。电流施加到嵌入硅片表面的电极上，会导致声波表面波在晶片上传播。嵌入晶片中的第二组电极(换能器)可以将这些波转换回电流。当电极和换能器连接到适当的电路时，这种表面波可以不断地由电路产生和放大。如果在同一晶片的表面嵌入第二组电极和换能器，就可以产生第二组波。这两个电路可以进行调谐，以便波以完全同步的方式振荡。通过覆盖晶片的一半(使用一对电极和换能器)，并将另一半暴露在环境中，可以构建非常灵敏的探测器。如果后半部分的表面受到干扰(例如，由于有害生物制剂的沉积)，同步被打破，这种情况可以由连接到电路的计算机立即检测到。这项研究的目标之一是为理解这些设备的行为提供数学理论。

The project encompasses the application of mathematics to four important physical problems: (1) the development of optimal protocols for the removal of glycerol from frozen blood; (2) the development of optimal strategies for bending and stretching one shape into another (for example, bending metal plates into curved structural surfaces or morphing in digital imaging); (3) the stability of water wave patterns for flow over plates; and (4) the modern physics two-body problem (that is, the theory of motion for two bodies whose motions produce waves, which travel with finite speeds, that influence the motion of the other body in the system). Glycerol must be added to whole blood to prevent cell damage when it is frozen and must be removed before the blood is transfused. Current removal strategies require procedures that are too lengthy for emergencies or military applications. Using mathematical modeling (which includes models that couple fluid flow, convection diffusion, and transport across semipermeable membranes), optimization theory and numerical methods, the project will scale up laboratory tested removal protocols for medical use. The research on optimal morphing uses differential geometry and elasticity theory to obtain cost functionals for minimal strain morphing. Minima over admissible sets of deformations (diffeomorphisms between compact hypersurfaces) will be proved to exist using the calculus of variations, and numerical methods will be developed to approximate these minima. The existence and stability of surface waves produced by flow over flat plates will be proved using fluid dynamical modeling and the theory of free boundary problems. In addition, the mathematical analysis will be complemented with numerical simulations of a physical system. Mathematical models of two-body interaction, where forces propagate with finite speeds, will be derived. Such models consist of systems of hyperbolic partial differential equations coupled with ordinary differential equations, or hybrid systems of differential equations and delay equations. The well-posedness and qualitative dynamics of the two types of models will be determined and models of this type will be applied to understand synchronization phenomena for acoustical surface wave sensors.brbrThe research project has two main purposes: the application of mathematics to advance understanding in four fields of physical science and the training of four Ph.D. students in these important areas of applied mathematics. The first project has direct applications in the health sciences and medicine. Living cells, for example blood cells, can be preserved for long periods of time by freezing; and, they can later be thawed for use in transfusions. To freeze and thaw living cells requires a complex technology including the addition of chemicals called cryoprotectants to the insides of the cells so they will not be damaged during this process. These chemicals must be removed before blood is transfused. Methods are available for doing this, but current methods require a long time to implement. Thus, frozen blood is not available for emergency use (for example, in military conflicts or natural disasters). Fast removal of cryoprotectants has been proved to be possible at the level of single cells; the purpose of the research in to design (optimal) methods that can be used for fast removal on the larger scale necessary for practical application in medicine. The second project has applications in materials science. A sheet of metal is to be bent and stretched into a desired shape. The desired shape can be achieved by many different deformation processes (for example, the sheet can first be stretched then bent or it can be stretched in one direction while being bent in another, among many other possibilities). Which method uses the least energy? This problem and others of the same type will be solved using mathematical analysis. The third project will determine the stability of the surface waves often formed by a fluid (for example water) moving over a plate. Surface waves are known to exist and have a particular shape. Also, experiments show that when the waves are disturbed, they return rapidly to that particular shape. But, the mechanism for this remarkable stability property is not known at present. The research project will determine the reason for the observed stability. The fourth research project is concerned with the theory of two-body interactions in physics and engineering. A practical example occurs in applied acoustics. Electrical currents applied to electrodes embedded in the surface of a silicon wafer causes acoustic surface waves to propagate across the wafer. A second set of electrodes (transducers), embedded in the wafer, can transform these waves back to electrical currents. When the electrodes and transducers are connected to an appropriate electrical circuit, such surface waves can be constantly produced and amplified by the circuit. If a second set of electrodes and transducers are embedded in the surface on the same wafer, a second set of waves can be produced. The two circuits can be tuned so that the waves oscillate in perfect synchrony. A very sensitive detector can be constructed by covering half of the wafer (with one pair of electrodes and transducers) and exposing the other half to the environment. If the surface of the second half is disturbed (for example, by the deposition of a hazardous biological agent) the synchrony is broken and this occurrence can be instantly detected by a computer connected to the circuits. A goal of this research is to provide the mathematical theory for understanding the behavior of these devices.